Statistics Education

Question 1

What is “Informal Statistical Inference” (ISI)?

Answer 1

ISI is based on generalizing beyond the given data, expressing uncertainty with a probabilistic language, and using data as evidence for these generalizations (Makar and Rubin, 2009, 2017)

Question 2

What is Informal Inferential Reasoning? (IIR)

Answer 2

The reasoning process leading to making ISI’s (Informal Statistical Inference). IIR refers to the cognitive activities involved in informally formulating generalizations (e.g. conclusions, predictions) about “some wider universe” from random samples, using various statistical tools such as: sample size, sampling variability, controlling for bias, uncertainty and properties of data aggregate (Rubin et al. 2006)

Question 3

Bivariate relations are characterized by…

Answer 3

Bivariate relations are characterized by the variability of each of the variables; the pattern of a relation, the shape of the relationship
in terms of linearity, clusters and outliers; and the existence, direction and strength
of a trend (Watkins et al. 2004).

Question 4

What does statistical covariation relate to?

Answer 4

Statistical covariation relates to the correspondence of variation of two variables that vary along numerical scales (Mortimer 2004)

Question 5

How is reasoning with covariation defined?

Answer 5

Reasoning with covariation is defined as the cognitive activities involved in coordinating, explaining and generalizing two varying
quantities while attending to the ways in which they change in relation to each other
(Carlson et al. 2002).

Question 6

What is meant by a “covariation approach”?

Answer 6

A covariation approach in this context entails being able to move between values of one variable and coordinating this shift with movement between corresponding values of another variable. Such an approach plays an important role in students’ understanding, rep-
resenting and interpreting of the rate of change, and its properties in graphs (Carlson
et al. 2002). The approach can also lead to reasoning about the algebraic representa-
tion of a function (Confrey and Smith 1994).

Question 7

What are the four levels of verbal and numerical graph interpretation?

Answer 7

Nonstatistical, single aspect, inadequate covariation, and appropriate covariation

Question 8

What do Nonstatistical responses relate to?

Answer 8

Nonstatistical responses relate to the context or to a few data points such as outliers or extreme values without addressing covariation.

Question 9

What do single aspects responses refer to?

Answer 9

Single aspects responses refer to a single data point or to one of the variables (usually the dependent), with no interpolating.

Question 10

What does inadequate covariation responses address?

Answer 10

Inadequate covariation responses address both variables but either relate to correspondence by comparing two or more points without generalizing to the whole data or to the population; Or variables are described without relating to the correspondence or by mentioning it incorrectly.

Question 11

What does appropriate covariation responses refer to?

Answer 11

Appropriate covariation responses refer to both variables and the correspondence correctly?

Question 12

Describe the challenges students face while reasoning with covariation.

Answer 12

Students tend to focus on:

1) Isolated data points rather than on the global data set and trend.
2) Single variable rather than the bivariate data
3) Expect a perfect correspondence between variables, without exception in data
(a deterministic approach)
4) Consider a relation between variables only if it is positive (the unidirectional misconception).
5) Reject negative covariations when they are
contradictory to their prior beliefs.
6) Have a hard time distinguishing between arbitrary and structural covariation (Batanero et al. 1997; Ben-Zvi and Arcavi 2001; Moritz
2004).

Question 13

What is aggregate reasoning?

Answer 13

Aggregate reasoning is a global view of data that tends to aggregate features of data sets and their propensities. (Ben-Zvi and Arcavi 2001; Shaughnessy 2007)
A data set is considered as a whole, emergent properties of the whole are different than properties of the individual cases.

Question 14

What are two important aggregate properties of a data set?

Answer 14

Two important aggregate properties are the distinction between signal and noise and the recognition and diagnosis of various types and sources of variability. (Ruben et al. 2006)

Question 15

What are 3 key aggregate aspects of a distribution?

Answer 15

Aggregate aspects of distribution:
Shape, Spread, Concentration (Makes me think of how I spread butter and jam on toast compared to how Beth does! Mine, random, clumpy, and tending to be all located in the middle. Beth’s smooth, evenly thick, no concentration in any one area.)

1) General shape,
2) How spread out the cases are
3) Where the cases tend to be concentrated within the distribution
(Bakker and Gravemeijer 2004; Konold et al. 2015).

Question 16

What should students search for when reasoning with data?

Answer 16

Signals in the variability. (Signals are patterns which are not caused by noise.)
Potential sources for the signals.

Questions that can promote this search:
What patterns do we see?
Do we think these patterns are just noise, or maybe something else?
What might be going on in the context that gives rise to these patterns?

Question 17

Describe informal and formal aspects regarding reasoning with variability.

Answer 17

It begins with the recognition that data vary! For example, a fair coin an equal chance to land on heads or tails. However, when we flip a coin 10 times, it is very likely that we will not get 5 heads and 5 tails. We will get some variation from this.

This leads us to want to quantify the variation in the data. How far from the theoretical expectation (5 heads and 5 tails) do we need in order to conclude that the coin is not fair? Ten heads and no tails? Nine and one? Eight and two?

This leads us into wanting to compare our coin to what we would see from a coin we know is fair.

Formal measures and methods then arise. Examples include measures of center (mean, median, mode), measures of spread (mean deviation, standard deviation…), the shape of a distribution, formulating a null hypothesis and an alternative hypothesis, significance levels, and the logic related to rejecting, or not rejecting, the null hypothesis.

Question 18

What does a conceptual understanding of variability include?

Answer 18

A conceptual understanding of variability includes: (a) developing intuitive ideas
about variability (e.g., repeated measurement on the same characteristic are variable);
(b) the ability to describe and represent variability (e.g., the role of different represen-
tation of a data set in revealing different aspects of variability, the representativeness
of spread measurements); (c) using variability to make comparisons; (d) recognizing
variability in special types of distributions (e.g., the role of the variability of both
variables’ distributions to a bivariate data distribution) ; (e) identifying patterns of
variability in fitting models; (f) using variability to predict random samples or outcomes; and (g) considering variability as part of statistical thinking (Garfield and
Ben-Zvi 2005).

Question 19

Define modeling.

Answer 19

Modelling is defined as simplifying or grasping the essentials of a static or dynamic
situation within a rich and dynamic context (Freudenthal 1991).

Question 20

Stages of modeling

Answer 20

Modelling can be perceived as interrelating processes in which the role of the model is changing as thinking progresses. At the first process, a model emerges as a “model of” informal reasoning and develops into a “model for” more formal reasoning. At the second
process, a new view of a concept emerges along the transition from “model of” to
“model for”. Such view can be perceived as formal in relation to the initial disposi-
tion toward this concept. These two processes are accompanied by a third one—the
shaping of a model as a series of signs that specifies the previous reasoning process
(Gravemeijer 1999).

Question 21

What are reasons why statistics is a challenging subject to learn and teach?

Answer 21

1) Complex, difficult, and/or counterintuitive.
2) Difficulty with the underlying mathematics— fractions, decimals, proportional reasoning, and algebraic formulas.
3) The contexts of many statistical problems may mislead students.
4) Students equals statistics with mathematics and expect the focus to be on numbers, computations, formulas, and only one right answer.

Firstly, many statistical ideas and rules are complex, difficult, and/or
counterintuitive. It is therefore difficult to motivate students to engage in the hard
work of learning statistics. Secondly, many students have difficulty with the
underlying mathematics (such as fractions, decimals, proportional reasoning, and
algebraic formulas) and that interferes with learning the related statistical concepts. A
third reason is that the context in many statistical problems may mislead the students,
causing them to rely on their experiences and often faulty intuitions to produce an
answer, rather than select an appropriate statistical procedure and rely on data-based
evidence. Finally, students equate statistics with mathematics and expect the focus to
be on numbers, computations, formulas, and only one right answer. They are
uncomfortable with the messiness of data, the ideas of randomness and chance, the
different possible interpretations based on different assumptions, and the extensive
use of writing, collaboration and communication skills. This is also true of many
mathematics teachers who find themselves teaching statistics.
(Ben-Zvi)