‘Misconceptions when Learning Statistics at Primary Level’ - From articles

Article 1:

How pre-service teachers integrate knowledge of students' difficulties in understanding the concept of the arithmetic mean into their pedagogy.

Summary:

This article is about the pre-service teachers whose read some articles on students’ difficulties with the concept of mean and find out that students have two major misconceptions regarding the basic concept.

Misconceptions:

• Confusion of the concept of mean and median. Both mean and median are seen as a point of balance and fall in the middle area of the data set, students often see average as being the exact middle number( actually it is median).
• Confusion between mean and mode. Mode is always seen as a way of representing a majority of information and occurs frequently, while it likened to the term average in many areas of life such as people normally eat outside than at home. Students give definition such as normal, most common and most frequent when asked about the meaning of average.

Article 2:

Anticipating and Addressing Students' Misconception

Summary:

Students are given informal assignment to identify and correct the error because it is believed that they can enhance problem-solving skill and assess the reasonableness of their own answer.

Misconceptions:

• Creating a two-way tables from raw data. For instance, students are asked whether Carlos has seen a movie on the American Film Institute’s “top 100” list and whether his lab partner Hannah has also seen it. Suppose that Carlos reports back that he has seen 36 of the movies and Hannah has seen 27. Students came out with a table which showed the total of 200 movies. It does not allow us to calculate conditional proportions to see if the two variables ({has Hannah seen the movie?} and {has Carlos seen the movie?}) are related. Therefore, we need one variable at column variable and one variable at row variable.
• Representing the naked number rather than the unit. For instance, given a table with two variable: Number of Inversion and Tally Count. Students are asked about the median of the data set. The median fall under 0 numbers of inversion and 84 tally count. Students represent the answer as 0 rather than saying the unit which is supposed to be 72 as the total is 144. 

Article 3:

Conceptions and misconceptions of average: A comparative study between teachers and students.

The concept of average is intimately related to the comprehension of the properties which according to Strauss and Bichler (1988) are:

1. the average is located between the extreme values (minimum value ≤ average ≤ maximum value);
2. the sum of the deviation from the average is zero (Σ(Xi – average)=0);
3. the average is influenced by each and by all the values (average = ΣXi/n);
4. the average does not necessarily coincide with one of the values which are composed by it.
5. the average may be a number which does not have a correspondent in the physical reality (for example, the average number of children per couple can be 2.3);
6. the calculation of average takes into consideration all the values including the negative and zero;
7. the average is a representative value of the data from which has been calculated. In spatial terms, the average is the value which is closer to all the values.

Misconceptions regarding average:

• confused with the average and the sum of the values
• the average got confused with the maximum value of the data, which is related to the lack of comprehension of property of average
• the average has to coincide at least with one of the values.

Article 4:

STATISTICAL MISCONCEPTIONS

Summary:

The misconception: There are three different measures of central tendency: the mean, the median, and the mode.

Examples of misconceptions in the understanding of central tendency:

• The central tendency is the tendency of the observations to accumulate at a particular value or in a particular category.The three ways of describing this phenomenon are mean, median, and mode.
• There are three measures of central tendency: mean, median, and mode.
• There are three kinds of average: the mean, the median, and the mode.

Correcting the misconception:

• It is best to think of the various kinds of central tendency indices as falling into three categories based on the computational procedures one uses to summarize the data.
• One category deals with means, with techniques put into this category if scores are added together and then divided by the number of scores that are summed. In the first category (means), we obviously find the arithmetic mean. However, other entries in this category include the geometric mean, harmonic mean, trimmed mean, winsorized mean, midmean, and quadratic mean.*
• The second category involves different kinds of medians, with various techniques grouped here if the goal is to find some sort of midpoint. In the second category (medians), we of course find the traditional median (which is equivalent to Q2, the 50th percentile). Three other kinds of central tendency also belong in this category: midrange, midhinge, and trimean.*
• The third category contains different kinds of modes, with these techniques focused on the frequency with which scores appear in the data. The third category of central tendency indices involves modes. Here, we find the traditional notion of the mode: the most frequently occurring score in the data set. In addition, three additional kinds of modes exist: minor mode, crude mode, and refined mode.

3. What we have learnt from these:

We realized how often we are not aware of our misconceptions about statistics especially in about central tendency as we used to believe that they are not important and cannot help much in real life situations. We put little emphasis on learning the meaning of central tendency except for learning about their formulas and how to identify them from a graph or a set of data. From these readings, we learnt the importance of correcting my misconceptions about statistics as my misconceptions would be transferred to my students if we don't deal with them right now. For instance, we used to believe that the three measures for central tendency are mean, median, and mode. However, after reading the article about central tendency we realized that mean, median, and mode are actually central tendency indices that can be categorized into three based on the computational procedures one uses to summarize the data. Under each category, there are in fact other types of means, modes, and medians that may not be applicable to primary level. Nonetheless, knowing this prevents us from teaching the kids the wrong conception about central tendency. In addition, we also learned the importance of finding these central tendency indices as it helps us to summarize data which help in comprehending and therefore making the data more useful to us when we make decisions. In short, we as teacher have to make the students realize the utilitarian value of statistics rather than a long list of list of terms to memorize and complex calculation to complete, while at the same time increase their statistical literacy and reasoning through project work, laboratory work, group problem-solving and discussion.

4. Ways to solve the misconceptions on AVERAGE & MEDIAN:

Confusion of the concept of mean and median. Both mean and median are seen as a point of balance and fall in the middle area of the data set, students often see average as being the exact middle number( actually it is median).

• Using the language properly

-Be careful when referring to average in classroom lesson and when engaging students in dialogue about median.

-This can help children to separate the two concepts and prevent confusion.
--Provide a simple and sweet definition to them. For instance, 

median= middle number, arrange in ascending order

mean= average number, plus and divide 

mode=most commonly occurring number

• Develop conceptual understanding

-Create a solid conceptual understanding. Such as, having students to hold paper with different numbers and stand in a line, then ask them to identify the mean, mode and median. It makes more sense to them when they are doing it and teachers can intervene immediately when there is a confusion.

The average has to coincide at least with one of the values.

• Giving example

-Tell students that there are instances where the average will  be the same as one of the values. However, it does not mean that the average has to coincide with one of it.
-Give students example such as: 

• 3+4+5=12, 12/3=4, so the average is 4 but it is just a coincidence. 
• 1+2+3+4+5+6=21, 21/6=3.5, so the average is not the same with any of the values. 

Our viewpoints

Our viewpoints

After reading articles about statistical thinking, we have summarised what we think about statistical thinking in a diagram as attached below.

This is a simple diagram that summarizes what we think about statistical thinking and the examples of what these thinking would look like.

A simple definition of statistical thinking that we both agreed upon is : the ability to generate possible questions, collect useful data, organize and representing data, analyse and make sound and logical decision by taking factors (such as availability of resources, ethical concerns) into consideration.

How do teachers move students up to the next level?

Generate questions:

Lead students to some questions to think about the problem and help them to  generate a statistical question. Then make sure their question can answer the problem they want to know. For example,

• how do we go about answering this question?
• what do we need to know?
• how will we find the information that we need?
• what will we do with the information that we collect?
• who will find this information useful?
• is this information relevant to the problem

Collecting data:

After identifying what they need to know, teachers can facilitate students to collect relevant data with the most appropriate method. For instance, whether the data collected can be done through an

• online survey
• Interviews
• Questionnaires
• registry (for a related event) etc.

Teachers facilitate students in choosing the most appropriate and effective way of collecting relevant data by having discussions and consultation during a project.

Organising and representing data:

To organise data collected and summarise them is not as easy as it seems especially when the collected data is massive. Teacher can help students to master this process by giving some examples of organising data. For instance, making a

• Table with different aspects
• List

In addition, teachers can also teach the way to find average, mean and mode as an effort to organise data.

Then, students will need to choose the best way to represent these organised data using any visual display. This is where students need to choose among

• A bar chart
• Line graph
• Pie chart
• Histogram

To effectively represent their data. Teachers can show students the characteristics of each type of graph in order to aid students in selecting the best representation.

Analyse data:

Based on the data represented, students should be prompt to analyse the data with the aim of finding a solution to the generated question. For instance, if the problem was about how to solve food waste in the school canteen, based on the data represented students should be prompted to make conclusions such as “most students have their lunch in the school canteen on Fridays”. In the effort of prompting, teachers can ask questions such as:

• What can you conclude from the graph?
• What does the number of students having lunch at school canteen on Mondays tell you?
• What is the pattern of the data?
• What does the pattern tells you? (e.g. most students will only buy food from canteen on Fridays.)
• What do you think can be the reason behind this pattern?

Making decision:

This is critical in statistical thinking because the decision made by the students would determine whether the problem will be solved and would affect the outcome greatly. Teachers can help students in making a sound decision by having them:

• Search information from the internet or clarify with teachers or parents when having doubts
• Apply their knowledge, for instance, if students were to solve the food waste issue, they can then apply what they have learnt from moral or science such as how to tackle the food waste.
• Think of the limitations and consequences of the decision.
• Consider different aspects such as if the canteen staff can accept the decision made by the students and if this is applicable for all the students because there will be students who eat more than the others. 
• Always have a backup plan. 

All in all, we both agreed that although the frameworks suggested by statisticians may be useful guidelines for teachers in teaching statistics such as carrying out learning activities to encourage students to think statistically, in a real classroom however, teachers will need to consider other factors that will affect the effectiveness of teaching and learning to decide on how they will deliver a statistics lesson so that students will all benefit from the lesson. For instance, teachers need to consider students' prior knowledge, interest, and readiness before deciding what should be done to facilitate students in learning statistics in a meaningful way.

Educators Viewpoints

Student's Individual and Collective Statistical Thinking

In an effort to identify statistical thinking framework, educators from Illinois State University have identified four key statistical processes: describing, organizing, representing, and analyzing and interpreting data. They based their framework on the assumption that primary school students would exhibit four levels of statistical thinking that were described as idiosyncratic, transitional, quantitative and analytic.

Key processes

The first process is to describe data or 'reading the data', which involves extracting information explicitly stated in the data displayed, recognizing graphical conventions, and making connections between context and data. For example, questions such as 'what does the graph tell you?' requires this thinking process.

The second process which is to organize and reduce data requires mental actions such as ordering, grouping, and summarizing data. One example of questions that require this thinking is 'what is the average number of students in the school?'

Representing data incorporates constructing visual displays that require different organizations of data. For instance, students would use this thinking process when asked to compare the number of sold ice cream to the number of sold salad bowl in a school canteen. Students need to figure out which type of graph will best show this comparison.

Students who are able to recognize patterns and trends in the data presented and make inferences and predictions from the data practice the final statistical thinking process, that is analyzing and interpreting data.

Thinking Levels

Based on the respondents answers, the below conclusions are made about how students act at different levels of statistical thinking.

Level 1: Idiosyncratic. At this level, students are limited to idiosyncratic reasoning that was often unrelated to the data presented. They often focus on their own personal data banks that may not be relevant to the given data.

Level 2: Transitional. Level 2 thinkers begin to realize the importance  of quantitative thinking and use numbers to invent measures for center (median/ mean of data) and spread (range of data). Their perspective on data is generally single-minded and there are seldom connections between representations or analyses of the data to its context.

Level 3: Quantitative. Students who are able to think at this level use quantitative reasoning consistently as the basis for statistical judgments and begin to form valid conceptions of center and spread. Students at this level are aware of both the context and the data but connections between the both are seldom made.

Level 4: Analytical. Students who exhibit this thinking use more analytical approaches in exploring data and can make connections between context and the data. They are also able to adopt both a macro and micro view of the data.

These are some examples collected from responses from interviewees that show the different responses at different level of statistical thinking.

Other Educators Viewpoints

Beth Chance& Allan Rossman (Professor of Statistics at Cal Poly and San Luis Obispo) suggested that statistical thinking involves careful design of a study to collect meaningful data to answer a focused research question, detailed analysis of patterns in the data, and drawing conclusions that go beyond the observed data.

Joel B. Greenhouse suggested that good statistical thinking requires a nontrivial understanding of the real-world problem and the population for whom the research question is relevant. It involves judgments such as those about the relevance and representativeness of the data, about whether the underlying model assumptions are valid for the data at hand and about causality and the role of confounding variables as possible alternative explanations for observed results. Finally, an essential component of good statistical thinking is the ability to interpret and communicate the results of a statistical analysis so non-statisticians can understand the findings.

Beth Chance (2002) also suggested statistical thinking is the ability to see the process as a whole (with iteration), including “why,” to understand the relationship and meaning of variation in this process, to have the ability to explore data in ways beyond what has been prescribed in texts, and to generate new questions beyond those asked by the principal investigator.

Statisticians Viewpoints

The Four Framework for Statistical Thinking in Empirical Enquiry

• This framework in figure 1 is built based on responses from statisticians on how one thinks statistically.
• It seeks to organise some of the elements of statistical thinking during data-based enquiry.
• In short, the thinking operates in all four dimensions at once.
• For instance, the thinker could be categorised as currently being in the planning stage of the Investigative Cycle (Dimension 1), dealing with some aspect of variation in Dimension 2 (Types of Thinking) by criticising a tentative plan in Dimension 3 (Interrogative Cycle) driven by scepticism in Dimension 4 (Dispositions).
• This pattern of thinking is not peculiar to statisticians, but the quality of thinking can be improved by gaining more statistical knowledge.

The four dimensional framework built by the authors of Statistical Thinking in Empirical Enquiry.

Dimension One: The Investigative Cycle

• The first dimension in Fig. 1(a) shows how one acts and thinks during a statistical investigation.
• This cycle is concerned with abstracting and solving a statistical problem grounded in a larger "real" problem which has the intention to improve current situation.
• A knowledge-based solution to the real problem requires better understanding of how a system works.
• For instance, when students are given a problem such as how to resolve the issue of wasted food in the school canteen, they will need to first define the problem such as how much food is wasted everyday by learning more about the problem. Then they would need to plan and analyse the issue based on what they learned. The cycle goes on with data collection such as finding out how many students actually buy their lunch from the school canteen, and analysing the data collected.  Then students will need to come up with a solution to solve this issue such as preparing lesser food on some days where most students would bring their own lunch or preparing more appetizing meals.

Dimension Two: Types of Thinking 

There are two major types of thinking: general types of thinking that are common to all problem solving and the thinking that are foundations in statistical thinking.

General types of thinking

Strategic thinking 

• Strategic thinking aimed at deciding upon what and how we will do it. For instance, planning how to solve a problem and anticipating problems to avoid them.
• It is important to have an awareness of the constraints one has while working to solve an issue. For example, being aware that one's perception will influence how one approach an issue which will then desensitizing some important information.
• By challenging our own perceptions during group discussions can remove an obstacles and lead to new insights.

Modelling

• Simplify the information and construct a model to represent the information
• We build statistical models to gain insights from this information ("interpret") which feed back into the mental model.
• "Statistical models" here is more general than something like logistic regression. It refers to all of our statistical conceptions of the problem that influence how we collect data about the system and analyse it.

Applying techniques

• Problem solving technique in math is to map a new problem onto a problem that has already been solved so that the previously devised solution can be applied or adapted.
• To use statistics, we first recognize elements of our context that can be usefully mapped onto a model (a process of abstraction from the particular to the generic), operate within that model, and then we map the results back to context (from the generic to the particular).

Fundamental statistical thinking

Recognition of the need for data

• The recognition of the inadequacies of personal experiences and anecdotal evidence leading to a desire to base decisions on deliberately collected data is a statistical impulse.

Transnumeration

• Think of a new way to represent data and to enhance or generate understanding.

Consideration of Variation

• Variation is an observable reality. It is present everywhere and in everything. Variability affects all aspects of life and everything we observe. Nothing is the same.
• It is variation that makes the results of actions unpredictable, that makes questions of cause and effect difficult to resolve, that makes it hard to uncover mechanisms. Eg: change the pattern of variation to something more desirable(reduce the accident rate)

Reasoning with statistical models

• The main contribution of the discipline of statistics to thinking has been its own distinctive set of models, or frameworks, for thinking about certain aspects of investigation in a generic way.
• Modelling tools aid in discovering valuable generic lessons through investigative processes.

Integrating the statistical and contextual

• One cannot be indulge in statistical thinking without some context knowledge.
• In order to arrive at a meaningful result, one has to make connections between existing context-knowledge and the results of analyses.

Dimension three: The Interrogative Cycle

The Interrogative Cycle illustrated in Fig. 1(c) is a generic thinking process in constant use in statistical problem solving. There are different components in this cycle:

Generate

• Think of the possibilities which come from the context, the data or statistical knowledge and apply to the present problem, or may be registered for future investigation

Seek

• Internal seeking which we observe people thinking and digging their memory for relevant knowledge
• External seeking such as obtaining information and ideas from sources outside the individual or team.
• Reading relevant literature.

Interpret

• Connecting new ideas and information with our existing mental models and enlarging our mental models to make connection.

Criticize

• Checking for internal consistency and against reference points (arguing with ourselves, weighing up against our context and statistical knowledge, against the constraints we are working under, and anticipate problems that are consequences of particular choices.) or against external reference points such as: (talk to clients, colleagues, experts, "workers in the system", available literature and other data sources).

Judge

• Make judgment of our decision whether what we do, what we ignore, what we want to research further.

Dimension four: Dispositions

This dimension describes personal qualities categorized in Fig. 1(d) which affect entry into a thinking mode. These elements are observed in the context of statistical problem solving.

Curiosity and Awareness

• Statistician Peter Mullins stressed the importance of "noticing variation and wondering why" for generating ideas for improving processes and service provision.
• This lead to engagement.

Engagement

• Background knowledge helps-it is hard to be interested in something one knows nothing about.

Imagination

• Imagination is viewing a situation from different perspectives, and generating possible explanations or confounding explanations for phenomena and features of data.

Scepticism

• Scepticism involves actions such as looking out for logical and factual flaws when receiving new ideas and information.
• Scepticism here was basically targeted towards, "Are the conclusions reached justified?". For example, "worry questions" concerning the appropriateness of the measurements taken, the appropriateness of the study design, the quality of the data, the suitability of the method of analysis, and whether the conclusions are really supported by the data.

Being logical

• Being logical ensure a valid conclusion. To be useful, scepticism must be supported by an ability to reason from assumptions or information to implications that can be checked against data.

A propensity to seek deeper meaning

• Always be prepared to dig a little deeper into issues by being open to new ideas as it helps to register and consider new ideas and information that conflict with our own assumptions and perseverance is self-evident.