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Contexts for Highlighting Signal and Noise


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During the past several years, we have conducted a number of instructional interventions with students aged 12 – 14 with the objective of helping students develop a foundation for statistical thinking, including the making of informal inferences from data. Central to this work has been the consideration of how different types of data influence the relative difficulty of viewing data from a statistical perspective. We claim that the data most students encounter in introductions to data analysis—data that come from different individuals—are in fact among the hardest type of data to view from a statistical perspective. In the activities we have been researching, data result from either repeated measurements or a repeatable production process, contexts which we claim make it relatively easier for students to view the data as an aggregate with signal-and-noise components.
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Chapter 18
Contexts for Highlighting Signal and Noise
Clifford Konold
University of Massachusetts Amherst
Anthony Harradine
Prince Alfred College
Abstract During the past several years, we have conducted a number of instruc-
tional interventions with students aged 12 – 14 with the objective of helping stu-
dents develop a foundation for statistical thinking, including the making of infor-
mal inferences from data. Central to this work has been the consideration of how
different types of data influence the relative difficulty of viewing data from a sta-
tistical perspective. We claim that the data most students encounter in introduc-
tions to data analysis—data that come from different individuals—are in fact
among the hardest type of data to view from a statistical perspective. In the activi-
ties we have been researching, data result from either repeated measurements or a
repeatable production process, contexts which we claim make it relatively easier
for students to view the data as an aggregate with signal-and-noise components.
1 Introduction
Suppose you wanted to introduce 12-year-old students to basic ideas in statistics
such as center and spread. Here are two short descriptions of classroom activities
involving the collection and analysis of data which you could use. Which option
would you select and why?
Activity 1. Students collect information about themselves including their gen-
der, height, and distance traveled to school. They explore questions such as
whether 12-year-old boys are taller than 12-year-old girls.
Activity 2. Each student measures the length of a table using two different
measurement instruments. They explore questions such as whether one instrument
gives a more accurate measure than the other.
Based on an informal analysis of published data activities, our guess would be
that most readers would prefer Activity 1. The overwhelming majority of activities
for all age levels are similar to Activity 1, in that students work with data from
contexts where the attributes they are dealing with, such as height, result from
Konold, C., & Harradine, A. (2014). Contexts for highlighting signal and
noise. In T. Wassong, D. Frischemeier, P. R. Fischer, R. Hochmuth, & P.
Bender (Eds.), Mit Werkzeugen Mathematik und Stochastik lernen: Using Tools
for Learning Mathematics and Statistics (pp. 237-250). Wiesbaden, Germany:
238 Clifford Konold, Anthony Harradine
what we will refer to as "natural" variability. Rarely do we encounter published
activities similar to Activity 2 that involve repeated measurements. As to the rea-
son that educators might give for their preference, we imagine many would regard
the question posed in Activity 1 as being of interest to students whereas the ques-
tion in Activity 2 seems rather boring. Would students really care about the length
of a table, let alone the characteristics of repeated measurements of it?
In this chapter we argue the pedagogical merits of using contexts such as Activ-
ity 2. During the past 8 years, we have conducted several instructional interven-
tions with students aged 12 – 14 as well as with teachers using contexts similar to
Activity 2 where our objective has been to establish a solid foundation for statisti-
cal thinking. In this article, we describe those objectives and explore the af-
fordances of different contexts in making those ideas visible to students and in
supporting classroom discourse about important aspects of those contexts.
2 Characteristics of Three Statistical Contexts
Fig. 18.1 Fruit sausages made by three different students. Ideally, sausages would all be the same
length and thickness and thus weigh the same.
In collaboration with Rich Lehrer and Leona Schauble, we have been pursuing a
speculation put forward in Konold and Pollatsek (2002)—that data from some
contexts are considerably easier than others to conceive of statistically as combi-
nations of signal and noise (see Konold and Lehrer 2008, for a review of some of
this research). In the context of repeated measurements, we have involved students
in measuring various lengths (e.g., their teacher's arm span, a table, the footprint
of a crime suspect). More recently we have tested manufacturing contexts includ-
ing packaging toothpicks, cutting paper fish to a desired length, and producing
Play-Doh "fruit sausages" of a specified size (see Figure 18.1).
There are many similarities between the contexts of manufacturing and repeat-
ed measurements but also some interesting differences (Konold and Lehrer 2008).
These differences have led us to consider whether manufacturing processes might
provide a more suitable context in which to involve young students. The main ad-
vantage is that it is clear in the manufacturing context why we are producing mul-
tiple objects—it is the nature of manufacturing. And we measure samples of them
for quality assurance. By contrast, in most repeated-measurements contexts (such
as determining the length of a person’s arm span), the reason for repeatedly meas-
Contexts for Highlighting Signal and Noise 239
uring is not as clear. In real life, we measure once or twice, and exercise care if the
measurement matters. For this reason, it might be rather challenging to motivate
students to repeatedly measure and to sustain their interest (though there appears
to be no lack of interest in Lehrer’s classrooms). Secondly, the outputs of a pro-
duction process are physical objects and not just values. Students can look at the
manufactured objects and note the variability even before they measure them (see
Figure 18.1). Later, when looking at measurement values, students can re-inspect
the physical objects, coordinating observations from graphs with features of the
real objects. Finally, that the data from the production process are individual ob-
jects makes the context closer, conceptually, to the context of natural variability.
Because of this similarity, it seems reasonable to expect that students could more
easily apply knowledge formed in the study of production processes to situations
involving natural variability.
In short, our claim is that in the contexts of repeated measurement and produc-
tion it is clear that 1) we are using our data to try to infer a single value (a signal)
and that 2) the variability in the observed values is a nuisance (noise) obscuring
the signal. By contrast, both the existence and nature of signal and noise in con-
texts of natural variability are difficult to conceive. When, for example, we sum-
marize with a single value the distribution of heights of a sample of adult males,
we find it rather hard to explain what we are trying to capture beyond perhaps the
population parameter if we are trying to make an inference. You can point to noth-
ing in the real world to which the mean of this sample of heights refers to, whereas
as the mean of a sample of measurements of a table refers to the actual length of
that table.
When we claim that it is easier to perceive signal and noise in the contexts of
data production and repeated measurement we do not mean to suggest that these
components can be directly perceived in data. The development and refinement of
these and other statistical constructs and perceptions are goals of our instruction.
Rather, our contention is that these contexts provide a more suitable beginning
point for developing statistical practices and perspectives in our students.
Our take on the nature of what it is students ideally learn and how they learn it
is consistent with the views of Bakker and Derry (2011). Their view is, in turn,
grounded in the philosophy of Robert Brandom (as cited in Bakker and Derry
2011) who argues that knowledge in a particular domain is more than a collection
of mental representations but rather comprises an interrelated web of ideas, skills,
and justifications. Consider, for example, using a mean of several measurements
to estimate the true length of a table. To do so, we probably have facility with the
algorithm along with knowledge about various characteristics of the mean. But
more fundamental to the use of the mean is this context are the reasons we give for
computing and using it, the hedges that we offer, our justification for removing
particular extreme values from our computation, the alternatives that we consid-
ered and our reasons for rejecting them, the explanations we give for our observa-
tions. Accordingly, as we describe the contexts and activities below, we pay par-
ticular attention to the classroom conversations that typically arise. It is from these
conversations about what students are doing, and why, that we believe a more so-
240 Clifford Konold, Anthony Harradine
phisticated "web" of statistical understandings emerge. Thus we can refine our
central claim as follows: by using activities involving repeated measurement and
production process we can motivate and shape the kinds of conversations among
students from which they can learn to perceive data as a combination of signal and
noise. Below, we elaborate our claim and analyze episodes from one of our class-
2.1 Repeated Measures and Manufacturing Contexts
When we repeatedly measure a feature of some object, we obtain a distribution of
values for that measure. Below, for example, are 19 measurements of the length of
a table made by different students using the same metal tape measure.
Fig. 18.2 Distribution of 19 measurements of the length of a table
Two of the questions we pose to students once they see the class data are: 1)
What is your best guess of the actual length of the table and 2) Why are the meas-
urements not all the same? These questions provoke some interesting responses.
Having observed one another measure the table, students can quickly generate
several explanations for the variability. Aspects they mention include the place-
ment of the tape's end hook, where at the other end a person sights, how parallel
the tape is placed to the table's edge, and how the measurement is rounded. Simi-
larly, for the various manufacturing contexts we have tried, students can provide
detailed descriptions of process components that produce variability. They can do
this because they have participated in the manufacturing or measurement process
and have observed their classmates doing the same and have noted differences in
how measurements are made or the product is created. In producing the fruit sau-
sages shown in Figure 18.1, for example, students used a device that extruded a
long strand of Play-Doh. But these strands had a different surface texture and were
of different width depending on how hard a student pressed down on the extruder's
handle. Additionally, students measured the strand to cut it into sections that were
supposed to be 5 cm long. However, the lengths of these pieces varied both due to
Contexts for Highlighting Signal and Noise 241
measurement error and also to differences in the cutting procedure. Finally, stu-
dents weighed the cut pieces and here again they could observe how carefully this
was done in different groups.
Thus it is not only the contexts themselves but the students' direct involvement
in making measurements or product that we believe allows them to fairly naturally
regard the resulting data as having two general components. One of these is the
signal, or fixed component. The true length of the table does not vary. Because
each measurement is performed on the same table, the length of the table is cap-
tured in some sense in each measurement. Similarly, in the manufacturing con-
texts there is a specific target—how much a fruit sausage should weigh or the
number of toothpicks in a package. We do not initially reveal this target number to
the students but rather tell them that if they carefully employ the prescribed pro-
duction procedure they will get very close to the target value. Again, since this
target does not change and each fruit sausage, for example, is made using the same
procedure, the signal from that process is present in the measured weight of both
an individual sausage and aggregate characteristics of many sausages.
The other component of data from the measurement and manufacturing con-
texts is its variability. This component is undesirable (hence the term noise) and is
caused by errors of various types. Because the students have themselves made
these measurements or manufactured the product, they know a lot about the fac-
tors which contribute to that variability. To highlight these features, we often have
them do measurements with two different tools or produce a product using two
different procedures, one of which tends to produce much more variability than
the other.
To summarize, in the repeated measurement and manufacturing contexts stu-
dents can generate reasonable hypotheses about causes of error, suggest ways to
reduce error, anticipate how the distribution of data obtained with a more accurate
measuring tool or more controllable process will compare with data obtained us-
ing a less accurate tool or process, and expect that the actual measurement or
manufacturing target lies somewhere in the middle of the distribution where the
observations tend to cluster.
A critical feature of these contexts is that the inference one needs to make from
data is clear. We want to know the length of the table or to determine whether our
process is creating product according to specification. This clarity of question and
purpose is often missing from activities with data and, as a result, the need for
making inferences and justifying how they are arrived at is never established (see
Makar and Rubin 2009). These conditions often exist when students are exploring
data from what we are calling the context of natural variability.
2.2 The Context of Natural Variability
This ability we described above to observe and even control the process from
which the data result is typically absent in the world of natural objects. Given data
242 Clifford Konold, Anthony Harradine
about the heights of boys and girls in their classroom, for example, students might
have filled out and observed others filling out a questionnaire from which the data
were collected. But they cannot observe firsthand the environmental and genetic
factors that influence individual difference in height. Especially for younger stu-
dents, the causes associated with these individual differences are rather mysteri-
ous. Furthermore, why should we consider all the heights of boys together in a
single distribution and summarize that distribution with something like a mean?
According to Stigler (1999, pp. 73-74) this conceptual difficulty was the reason
that it took as long as it did to apply statistical thinking to what we have called
natural objects:
The first conceptual barrier in the application of probability and statistical methods in the
physical sciences had been the combination of observations; so it was with the social
sciences. Before a set of observations, be they sightings of a star, readings on a pressure
gauge, or price ratios, could be combined to produce a single number, they had to be
grouped together as homogeneous, or their individual identities could not be submerged in
the overall result without loss of information. This proved to be particularly difficult in
the social sciences, where each observation brought with it a distinctive case history, an
individuality that set it apart in a way that star sightings or pressure readings were not. …
If it were felt necessary to take all (or even many) of these [distinct case histories] into
account, the reliability of the combined result collapsed and it became a mere curiosity,
carrying no weight in intellectual discourse. Others had combined … [the individual
cases], but they had not succeeded in investing the result with authority.
2.3 Comparing Critical Features of the Contexts
In Table 18.1 we summarize the critical features of data from these three contexts,
repeated measurements, manufacturing, and natural objects.
Consider first the types of processes that produce the objects in the three con-
texts. Both repeated measurement and manufacturing involve processes that, in
theory, we can observe; the processes involved in the creation of natural objects
are generally unobservable and complex. Consider next the nature of the variabil-
ity (or noise). In the repeated measurement and manufacturing contexts, variability
is undesirable. In these contexts, if it were possible, we would eliminate all varia-
bility. Indeed, because we are partially in control of these processes, we can work
to minimize variability. In the case of living natural objects, variation is a positive
feature, critical to species survival. If we refer to variability in these natural con-
texts as noise, we do so only in a metaphoric sense. Finally, consider the signal as
indicated by the mean, for example. In distributions of repeated measurements and
manufactured objects, the mean of a sample is an estimate of a real-world entity—
the length of a table or the target of the manufacturing process. To refer to means
as signals in the case of natural objects is, again, only metaphoric as they have no
specific real-world referents. Quetelet (1842), who was the first to apply statistics
like the mean to measurements on people, invented the imagery l'homme moyen
(average man) to stand in place of a concrete referent.
Contexts for Highlighting Signal and Noise 243
Table 18.1 Comparison of critical features of data from three different contexts
Context Process Noise Signal
3 Making Sausages: A Teaching Experiment
In April 2007 we conducted a weeklong teaching experiment with 15 students,
aged 13-14, at Prince Alfred College, an all-boys private school in Adelaide, Aus-
tralia. Our activity was centered around the production of “fruit sausages,” small
cylindrical pieces of Play-Doh that ideally were to measure 5 cm in length and 1
cm in diameter. If made according to these dimensions we determined that the
sausage would weigh 4.4 grams, but we did not reveal this target weight initially
to the students; we hoped that they could eventually infer it from analysis of their
data. On the first day of production, each student made five fruit sausages by
hand. Working in teams of three they then weighed the sausages using first a triple
beam balance and then a digital scale. Analysis of the data showed the weights to
be quite variable, which set the stage for introducing a pressing device that stu-
dents could use to squeeze out a 1 cm diameter length of material. Continuing to
work in teams, students made another batch of sausages using this device. Analy-
sis of these data culminated with students developing measures of center and vari-
ability to decide which team's product was 1) closest to specification and 2) was
least variable. We then asked them to anticipate what the data would look like if
we made 1000 fruit sausages. Finally, students ran and critiqued a model built in
TinkerPlots (Konold and Miller 2012) to simulate their manufacturing process to
test their hypothesis about what this distribution of 1000 would look like. Below,
244 Clifford Konold, Anthony Harradine
we analyze excerpts from the classroom conversation, highlighting aspects of the
context that we believed helped to support the development of ideas of signal and
3.1 Measurements on the Hand-Made Sausages
Looking at the data of their weight measurements sparked animated conversation
about the variability they observed. One student's data attracted particular atten-
tion as his sausages were very fat, weighing over twice as much as most other stu-
dents' product. Students concurred that these fat sausages would be extremely
popular with consumers. But representing management we reminded them of the
bottom line and the importance of keeping down costs. This analysis and discus-
sion motivated the need for a process that would produce more consistent results
closer to the product specifications.
In most of our activities we have students use at least two measurement meth-
ods and/or production processes so as two produce distributions of different char-
acteristics. Here, the student expected the measurements using the triple beam bal-
ance would be more variable. In fact, they were of the opinion that the digital
scale would measure nearly perfectly, and because it served our purposes we did
not try to disabuse them of this belief. By subtracting the weights obtained from
the digital scale from those from the triple beam, we created a new attribute (see
Figure 18.3) that the students could consider as the error in the weighings from the
triple beam. The color indicates the team from which the measurements came.
Fig. 18.3 The differences between the weights as determined on the digital scale and triple beam
balance. The sausage on the far right weighed 1.6 grams more on the triple-balance beam than on
the digital scale. Circles are colored according to the student team who made the measurements.
Looking at this graph, students quickly focused on the cluster of values on the
far right. To help the conversation, we separated the distribution into the different
teams (Group) and placed a reference line at 0 on the axis (Figure 18.4).
Contexts for Highlighting Signal and Noise 245
Fig. 18.4 This graph was made by pulling the groups out of the distribution shown in Figure 18.3
and adding a reference line to indicate where the values would be if a sausage weighed the same
on both scales.
Below is part of the classroom discussion prompted by the graph in Figure
18.4. Ted was a member of the Fruits group whose data is on the lower right of the
graph. CK is Konold.
Ted: We thought our triple-beam balance was out of whack by about 1 gram.
CK: So why do you think that?
Ted: We were over weighing it, so, because all of ours [the differences] were
consistently around 1.
CK: ... And how would it be out of whack by a---
Jay: The thing [adjustment mechanism] at the back.
Ted: Yeah, because you can adjust it.
CK: Did you guys not adjust it?
Ted: No. ... And then what we decided is that it was out of whack by 1 because all
our ---, we have a big clump around the one mark, and nothing else anywhere
CK: And what does it mean that you guys are all close here?
Ted: We’re pretty consistent weighers.
From this graph Ted makes an inference about the amount of bias in the weigh-
ings from their triple beam balance. His estimate of 1 gram is based on where their
difference values are "clumped" in the distribution. At this early stage in the multi-
day activity, we were not expecting students to characterize their data using for-
mal averages. Our hope was that, as Ted does here, students would begin to use
their informal notions of modal clump (Konold et al. 2002) as an indicator of cen-
ter and draw inferences from that about the location of an unknown but specific
value—a signal. In this case, that signal is the amount of bias introduced by their
failure to calibrate the scale before using it. As indicated by his final comment, by
this time Ted was interpreting variability in terms of consistency.
Students were also beginning to distinguish between consistency (limited
noise) and accuracy (being close to the true value) as indicate by the exchange be-
246 Clifford Konold, Anthony Harradine
low. This conversation was initiated by Harradine (AH) who directed the class's
attention to the data in Figure 18.4 from some of the other teams. These notions of
consistency and accuracy are, of course, interpretations of noise and signal. Data
from the contexts of production and repeated measurement lend themselves to
these interpretations of consistency and accuracy in a way that data from contexts
of natural variability usually do not.
AH: So, I want to know then, who are the Appleheads? So, if what they [Ted] said is
true, what’s the story with the Appleheads?
CK: So are they as consistent as measurers as you guys?
Keith: No, but they were the most accurate.
AH: So hang on, whoa. So what does it mean that they’re closer to 0?
Jamie: It means they adjusted their triple-beam balance closer to the 0 mark.
AH: Now these four other groups up here?
?: They’re all spread out.
AH: So who’s the Beta group? You guys. How do we now, how do we explain why
you guys were like this?
Jamie: … Because we adjusted the scales after I’d used it [Jamie’s data are on the far
left of the Beta group.]
AH: Ahhh, you adjusted the scale. ... So these have shifted, so you reckon if they…
Jamie: If they, if Ken and Keith hadn’t changed it, then they’d be kind of on top of each
other, because they wouldn’t have been…
AH: Oh, you reckon they were adjusted in between when you measured them?
Keith: Yeah, he measured it, and then I adjusted it to 0.
AH: And then you measured some more. Ah, so where would these be [if the scale
had been adjusted first]?
Jamie: Underneath the others, the big purple clump.
AH: These ones [on the right]?
Jamie: Yeah.
Explaining variability is one of the primary objectives of data analysis. Here
Keith is imagining their single distribution of weight differences as comprising
two groups—those made before and those made after the scale was recalibrated.
These students' experience of producing and then measuring the sausages provides
the critical source for these explanations. They can relate some of the variability in
their data to their own actions and thus offer explanations for the variability. With
prompting, Jamie could imagine transforming their data to remove the effects of
this manipulation, mentally moving one cluster of data into the other, effectively
removing the effect of the recalibration and reducing the overall variability.
3.2 Models of Measurement and Production
In most of our activities involving measurement error or production systems, we
conclude by having students build and/or run simulations of the situations they
have been investigating. One purpose of this modeling component is to explore
statistical aspects of the situations that are not possible with data from the real sit-
uation, mainly because with limited time, we cannot collect enough data. The
Contexts for Highlighting Signal and Noise 247
more important reason we include the modeling component is to objectify the sig-
nal-noise, statistical view of the situation. By building and working with models of
the situations, we hope to foster a more generalized view of signal-noise, one we
hope students can eventually apply to a wide variety of contexts, including con-
texts of natural variability.
Figure 18.5 shows the model we introduced to the students on the final day of
the teaching experiment, describing it as a model of the process for making sau-
sages using the extruder according to our specifications. By this point, we had in-
formed students that the ideal sausage would weigh 4.4 grams. This value is en-
tered in the counter object on the left of the sampler. The range and distribution of
values in the "variation" spinner are based roughly on the class's experience with
making and weighing sausages. To model the making of, for example, 30 sausag-
es, we hit run. The sampler first selects the value 4.4 from the counter object and
then selects a value for weighing error by activating the spinner on the right and
taking the value from the slice the spinner's arrow randomly lands in. These two
values, the target and the error, are sent to the data table on the right where they
are summed to give a final modeled weight. For the first simulated sausage, that
weight is 4.5.
Fig. 18.5 A model of the process of making fruit sausages with the extruder built in TinkerPlots.
The sampler in the upper left contains the target weight of 4.4 grams. A random value for error
from the target weight is selected from the variation spinner. On each trial, these two values are
selected and sent to the table on the right where they are added to get the value of Weight. The
shape of the distribution of values shown below is determined by the make up of the variation
spinner; its position along the axis is determined by the target value.
248 Clifford Konold, Anthony Harradine
After briefly explaining the model and then running it to make four simulated
sausages, we asked:
CK: So, what do you think?
?: Uh, good!
AH: Will it make data like our sausages? Is it a good model do you reckon?
Various: Yeah.
AH: Go on, Vern.
Vern: It’s a decent model, but, yeah, but it’s not exactly humanlike.
CK: It’s not humanlike? What do you mean?
Vern: As in humans usually do something wrong with theirs...
CK: Give me an example of a human kind of error that we wouldn’t get here.
Vern: Like, if someone just repeatedly smashed on the squisher [pressed hard on the
Play-Doh extruder]. That would make a different weight than, probably, one of
those [simulated sausages from the sampler].
Examining a model tends to surface a number of ideas students have about the
process. Vern proposed that while the model is "decent" it does not take into ac-
count the sort of systematic effects they had observed, for example that pressing
hard on the extruder produced thicker sausages that weighed more than those
made by pressing more gently. Wyatt added that the model did not allow for the
possibility of improvement over time. Note that he made his argument by describ-
ing not the overall weight but rather the error component of the weight, a way of
thinking that we were aiming to promote with the model.
Wyatt: Another way it’s not humanlike is as humans tend to do more, they tend to get
more consistent. This [the sampler] is just doing it random. So one time it might
do a 0 and then the next time you might do a 0.5, like adding on. But here …
CK: How do you know this one is random?
?: Because it is just a spinner.
Wyatt: Yeah, cause this is just a spinner thing. Well, I’m guessing that it’s random.
Probably would be....
Wyatt: No, it’s not humanlike, because, um, if you think about it, as you make more
sausages, like by hand, you’d get generally more consistent and be around the
same number more. But this, like just say had I made one sausage, sausage
number 100, at 4.4 in grams, then the next one 4.5, to be consistent. But then on
this [sampler model] you might have one that is 4.0 and then one at 4.05 or
something, 'cause it’s just random.
Ted: Aah, [I've got a] comment about it. It doesn’t take into account that each group
makes it slightly differently.
CK: Right.
Ted: Because, for example, the Fruits had theirs focusing around about 4, and the
Appleheads had theirs focusing around 4.4. And we’re assuming the factory
always makes them precisely on 4.4.
People have a strong inclination to offer explanations for events or trends even
when there is good evidence that nothing but chance is responsible. So at the same
time we encourage students to pose theories that account for trends, we also want
them to develop the ability and habit to question whether the patterns they notice
might have resulted from chance. Taking this idea into consideration is the prime
motivator for statistical inference. While it certainly is reasonable to expect that
makers of fruit sausages get better with practice, we have students both look at
Contexts for Highlighting Signal and Noise 249
their data for evidence of this and also run the model and see whether they can get
similar "trends." Thus we do not typically compute any probabilities, but find that
without many trials from the sampler, student develop the sense that some rather
stunning patterns can occur just by chance. For this purpose we have used the kind
of display seen in Figure 18.6 because it more directly depicts than does the
stacked dot plot the idea of a signal with noise scattering data around it randomly.
It is especially powerful (and entertaining) to watch it build up in real time.
Fig. 18.6 The weights of 15 simulated sausages displayed as a time series
4 Conclusion
We have argued that classroom activities that involve repeated measurements and
manufacturing are particularly suitable for introducing students to statistics. This
is primarily because in these contexts statistical properties of distributions, includ-
ing indicators of center and spread, refer to properties of actual objects. In these
contexts, it is clear to students that the objective is to infer these properties (such
as the length of a table or the target of the fruit-sausage process) from the data
they collect. Furthermore, in the activities we have tested and described, students
not only collect the data but exercise some amount of influence over them. Their
actions and decisions impact both the accuracy and the consistency of their data.
Observations they make during the data-collection phase provide a source of ex-
planations of the trends and variability in the data. These characteristics function
together to fuel conversation among students about the nature and meaning of
their data and the conclusions they can draw from them.
Acknowledgments This work was supported by grants ESI 0454754 and DRK-12 0918653
from the National Science Foundation. The views expressed are our own and do not necessarily
represent those of the Foundation.
250 Clifford Konold, Anthony Harradine
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... Researchers have often recommended that graphical representations be used to foster and build students' understanding of central tendency (e.g., Konold & Harradine, 2014;Leavy, Friel, & Mamer, 2009;Lehrer, Kim, & Jones, 2011). Bakker, Derry, and Konold (2006), however, note the difficulties students have understanding the characteristics of graphical representations for this purpose, in particular box plots. ...
... They then used the formal mean value to compare the distributions of the two types of fish. Konold and Harradine (2014) also reported students aged 12-14 identifying modal clumps as indicators of center before they used formal measures such as the mean when investigating a manufacturing process. ...
... Following the work of Bakker et al. (2006), recent studies have also been exploring students' understanding from broader perspectives. Complementing this are suggestions from Konold and Harradine (2014), who claim that repeated measurements and production processes are particularly fruitful contexts for introducing students to statistical ideas about variation and measures of center. Makar (2014) also illustrated how basing the development of informal inference within inquiry-based learning experiences can support young students to develop rich conceptions of central tendency. ...
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This chapter presents an overview of the Practice of Statistics focusing mainly on research at the school level. After introducing several frameworks for the practice, research is summarized in relation to posing and refining statistical questions for investigation, to planning for and collecting appropriate data, to analyzing data through visual representations, to analyzing data by summarizing them with specific measures, and to making decisions acknowledging uncertainty. The importance of combining these stages through complete investigations is then stressed both in terms of student learning and of the needs of teachers for implementation. The need for occasional backtracking is also acknowledged, and more research in relation to complete investigations is seen as a priority. Having considered the Practice of Statistics as an active engagement by learners, the chapter reviews presentations of the Big Ideas underlying the practice, with a call for research linking classroom investigations with the fundamental understanding of the Big Ideas. The chapter ends with a consideration of the place of statistical literacy in relation to the Practice of Statistics and the question of the responsibility of the school curriculum to provide understanding and proficiency in both.
... Nonetheless, no matter how "carefully" students measured and how assiduously they avoided "mistakes," they found although they could influence the magnitude of variability, they could not eliminate it (Lehrer, Kim, & Schauble, 2007). Konold and Harradine (2014) suggest that, in contexts of repeated measure and production, "… because we are in control of these processes, we can minimize variability" (p. 242). ...
... Modeling helped students appreciate this variability as a result of a signal of sample space and noise of chance departures from the signal. In further exploration of modeling signal and noise, students (ages 13 and 14) participated in a simulated manufacturing process during a weeklong teaching experiment (Konold & Harradine, 2014). They manufactured "fruit sausages" composed of Play-Doh with two methods of production. ...
This chapter synthesizes diverse research investigating the potential of inducting elementary grade children into the statistical practice of modeling variability in light of uncertainty. In doing so, we take a genetic perspective toward the development of knowledge, attempting to locate productive seeds of understandings of variability that can be cultivated during instruction in ways that expand students’ grasp of different aspects and sources of variability. To balance the complexity and tractability of this enterprise, we focus on a framework we refer to as data modeling. This framework suggests the inadvisability of piecewise approaches focusing narrowly on, for instance, computation of statistics, in favor of more systematic and cohesive involvement of children in practices of inquiring, visualizing, and measuring variability in service of informal inference. Modeling variability paves the way for children in the upper elementary grades to make informal inferences in light of probability structures. All of these practices can be elaborated and even transformed with new generations of digital technologies.
... To continue the development of visualization and measure of distribution, we introduce a new signal and noise process, based on contexts of production (Konold and Harradine 2014). For example, students attempt to manufacture "candies" of a standard size out of clay. ...
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Because science is a modeling enterprise, a key question for educators is: What kind of repertoire can initiate students into the practice of generating, revising, and critiquing models of the natural world? Based on our 20 years of work with teachers and students, we nominate variability as a set of connected key ideas that bridge mathematics and science and are fundamental for equipping youngsters for the posing and pursuit of questions about science. Accordingly, we describe a sequence for helping young students begin to reason productively about variability. Students first participate in random processes, such as repeated measure of a person’s outstretched arms, that generate variable outcomes. Importantly, these processes have readily discernable sources of variability, so that relations between alterations in processes and changes in the collection of outcomes can be easily established and interpreted by young students. Following these initial steps, students invent and critique ways of visualizing and measuring distributions of the outcomes of these processes. Visualization and measure of variability are then employed as conceptual supports for modeling chance variation in components of the processes. Ultimately, students reimagine samples and inference in ways that support reasoning about variability in natural systems.
... Statistical modeling has typically been a topic addressed in advanced courses, although its use from kindergarten through introductory statistics has been an emerging development (e.g., delMas, Garfield & Zieffler, 2014;English, 2012;Fielding-Wells & Makar, 2015;Konold & Harradine, 2014;Lehrer, Jones & Kim, 2014;Lehrer, Kim & Jones, 2011;Makar, 2016;Peters, 2011). Our literature review focused on research relevant to the context of our study: developing the model-based reasoning of primary school children. ...
Teaching from an informal statistical inference perspective can address the challenge of teaching statistics in a coherent way. We argue that activities that promote model-based reasoning address two additional challenges: providing a coherent sequence of topics and promoting the application of knowledge to novel situations. We take a models and modeling perspective as a framework for designing and implementing an instructional sequence of model development tasks focused on developing primary students' generalized models for drawing informal inferences when comparing two sets of data. This study was conducted with 26 Year 5 students (ages 10-11). Our study provides empirical evidence for how a modeling perspective can bring together lines of research that hold potential for the teaching and learning of inferential reasoning. © International Association for Statistical Education (IASE/ISI), November, 2017.
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As STEM education and STEM literacy become more prominent at the school level, the question arises as to what the beginning experiences should be that will create a foundation for students as they begin their journeys with respect to STEM. The research reported here is based on the premise that building statistical understanding and statistical literacy simultaneously with STEM understanding and STEM literacy will enhance the experiences in both areas. The fundamental concept underlying both statistics and STEM is variation. This paper hence reports on an activity created to introduce students in Year 3 to the concept of statistical variation with data in a STEM-related context where variation occurs in an easily measured and realistic fashion. The purpose was to build the foundational understanding for a longitudinal study using modelling with data to enhance STEM contexts in the primary school. Students’ capacities to appreciate the fundamental nature of statistical variation and use it for comparison in a STEM context was assessed through responses in student workbooks, to questions on an end-of-year survey, and in individual interviews. Results showed that students could take on the idea of variation and use it in explaining their experiences with the hands-on activity, although there was a range of sophistication in the way the understanding was expressed. (eprint available at
Some researchers advocate a statistical modeling approach to inference that draws on students’ intuitions about factors influencing phenomena and that requires students to build models. Such a modeling approach to inference became possible with the creation of TinkerPlots Sampler technology. However, little is known about what statistical modeling reasoning students need to acquire. Drawing and building on previous research, this study aims to uncover the statistical modeling reasoning students need to develop. A design-based research methodology employing Model Eliciting Activities was used. The focus of this paper is on two 11-year-old students as they engaged with a bag weight task using TinkerPlots. Findings indicate that these students seem to be developing the ability to build models, investigate and posit factors, consider variation and make decisions based on simulated data. From the analysis an initial statistical modeling framework is proposed. Implications of the findings are discussed.
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The idea of data as a mixture of signal and noise is perhaps the most fundamental concept in statistics. Research suggests, however, that current instruction is not helping students to develop this idea, and that though many students know, for example, how to compute means or medians, they do not know how to apply or interpret them. Part of the problem may be that the interpretations we often use to introduce data summaries, including viewing averages as typical scores or fair shares, provide a poor conceptual basis for using them to represent the entire group for purposes such as comparing one group to another. To explore the challenges of learning to think about data as signal and noise, we examine the "signal/noise" metaphor in the context of three different statistical processes: repeated measures, measuring individuals, and dichotomous events. On the basis of this analysis, we make several recommendations about research and instruction.
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Technologies of writing have long played a constitutive role in mathematical practice. Mathematical reasoning is shaped by systems of inscription (i.e., writing mathematics) and notation (i.e., specialized forms of written mathematics), and systems of inscription and notation arise in relation to the expressive qualities of mathematical reasoning. Despite this long history, something new is afoot: Digital technologies offer a significant expansion of the writing space. In this essay, we begin with a view of the developmental origins of the coordination of reasoning and writing, contrasting notational systems to more generic forms of inscription.Then, following in the footsteps of Jim Kaput, we portray dynamic notations as a fundamental alteration to the landscape of writing afforded by digital technology. We suggest how dynamic notations blend the digital character of notation with the analog qualities of inscription, resulting in a hybrid form that is potentially more productive than either form in isolation. We conclude with another proposition spurred by Jim Kaput: Digital technologies afford new forms of mathematics. We illustrate this proposition by describing children’s activities with TinkerPlots™ 2.0, a tool designed to help students organize and structure data, and to relate their understandings of chance to these patterns and structures in data.
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Informal inferential reasoning has shown some promise in developing students' deeper understanding of statistical processes. This paper presents a framework to think about three key principles of informal inference - generalizations 'beyond the data,' probabilistic language, and data as evidence. The authors use primary school classroom episodes and excerpts of interviews with the teachers to illustrate the framework and reiterate the importance of embedding statistical learning within the context of statistical inquiry. Implications for the teaching of more powerful statistical concepts at the primary school level are discussed.
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This theoretical paper relates recent interest in informal statistical inference (ISI) to the semantic theory termed inferentialism, a significant development in contemporary philosophy, which places inference at the heart of human knowing. This theory assists epistemological reflection on challenges in statistics education encountered when designing for the teaching or learning of ISI. We suggest that inferentialism can serve as a valuable theoretical resource for reform efforts that advocate ISI. To illustrate what it means to privilege an inferentialist approach to teaching statistics, we give examples from two sixth-grade classes (age 11) learning to draw informal statistical inferences while developing key concepts such as center, variation, distribution, and sample without losing sight of problem contexts.
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SUMMARY This paper examines ways in which coherent reasoning about key concepts such as variability, sampling, data, and distribution can be developed as part of statistics education. Instructional activities that could support such reasoning were developed through design research conducted with students in grades 7 and 8. Results are reported from a teaching experiment with grade 8 students that employed two instructional activities in order to learn more about their conceptual development. A "growing a sample" activity had students think about what happens to the graph when bigger samples are taken, followed by an activity requiring reasoning about shape of data. The results suggest that the instructional activities enable conceptual growth. Last, implications for teaching, assessment and research are discussed.
Examines the history of some of the concepts involved in bringing statistical argument "to the table" and some of the pitfalls that have been encountered. Topics from the collection of essays include statistics and social science, Galtonian ideas, 17th century explorers, and questions of discovery and standards. Some essays examine deep and subtle statistical ideas such as the aggregation and regression of paradoxes while others tell of the origin of the Average Man and the evaluation of fingerprints as a forerunner of the use of DNA in forensic science. Several of the essays are entirely nontechnical, yet all examine statistical ideas with an ironic eye for their essence and what their history can tell us for current disputes. (PsycINFO Database Record (c) 2012 APA, all rights reserved)
Students' use of modal clumps to summarize data
  • C Konold
  • A Robinson
  • K Khalil
  • A Pollatsek
  • A Well
  • R Wing
  • S Mayr
Konold, C., Robinson, A., Khalil, K., Pollatsek, A., Well, A., Wing, R., & Mayr, S. (2002). Students' use of modal clumps to summarize data. In B. Phillips (Ed.), Proceedings of the Sixth International Conference on the Teaching of Statistics [CD ROM]. Cape Town: International Statistical Institute.