Digging Deeper into … Representing and Interpreting Data (infants to second class)

For practical suggestions for families, and links to useful digital resources, to support children learning about the topic of data, please check out the following post: Dear Family, your Operation Maths Guide to Data

Data Analysis Process

Data analysis, whether at lower primary, upper primary, or even at a more specialised level of statistics, is essentially the same process:

• It starts with a question, that doesn’t have an obvious and/or immediate answer. Information is then collected relevant to the question.
• This collected information or data is represented in a structured way that makes it easier to read.
• This represented data is then examined and compared (interpreted) in such a way as to be able to make statements about what it reveals and, in turn, to possibly answer the initial question (if the question remains unanswered it may be necessary to re-start the process again, perhaps using different methods).

Thus, every data activity should start with a question, for example:

• What is the most common eye/hair colour in our class?
• Which fruit/colour do we prefer?
• How did we come to school today?
• What pets do we have?
• What age are most people in this room?
• What weather did we have in the past week/month?
• Where did we go on holidays?
• Where do our grandparents live?
• Which picture book should we read next?
• How can we sort out our shoes?
• Who is in school today?

When choosing a question it is worth appreciating that some questions might not lend themselves to rich answers. Take, for example, the first question above; once the data is collected, and represented, there is not that much scope for interpretation of results other than identifying the most common eye/hair colour and comparing the number of children with one colour as being more/less than another colour. However, other questions might lead to richer answers, with more possibilities to collect further information, to make predictions and to create connections with learning in other areas. Take, for example, the question above about travel; the children could be asked to suggest reasons for the results e.g. can they suggest why they think most children walked/came by car on the day in question, whether weather/season/distance from school was a factor and to suggest how the results might be different on another day/time of year. Even in a very simple way, the children are beginning to appreciate that data analysis has a purpose i.e. to collect, represent and interpret information, so as to answer a question.

Sets and Data

Data is very closely related to sorting and classifying sets:

• The initial question may focus on a particular set in the classroom e.g. identifying the most common/frequent occurring item in the set of farm animals, the set of buttons in our button box, the shoes that the children are wearing, the nature items collected on the walk etc
• Information is then collected by sorting and classifying the items in the original set according to the target attribute.
• This collections of items are represented in a structured way that makes it easier to compare e.g. items put in lines of same type, use cubes or drawings to represent the actual items.
• This represented data is interpreted to answer the question and to make other statements about  relationships e.g. which group has more, less etc

Thus sorting and classify activities should be viewed as potential springboards into data activities and it is important that the children realise that they can represent and compare the size of the sets within each sort by graphing them.

CPA Approach

Even as the children move into first and second classes, it is important that their data activities continue to follow a CPA approach:

Concrete: Continuing to use real objects initially to sort and classify ) e.g. the number of different colour crayons in a box, the different type of PE equipment in the hall , the different fruit we brought for lunch etc), progressing towards using unifix cubes, blocks, cuisinere rods etc to represent the same data. Indeed, the children themselves could be used at this stage; sort the children into groups according to eye colour, hair colour,  age etc and get them to organise themselves into lines that represent the same criterion. This is turn can be very useful for the children to realise that how they are lined up is crucial to being able to interpret the data easily and correctly. If you have visible tiles/markings as flooring on the classroom/hall/corridor, these can be used to organise the “data” accurately!

Pictorial: using multiple copies of identical images to make pictograms and/or using identical cut out squares/rectangles of paper on which the children draw an image that represents the data as it relates to them (e.g. how I traveled to school today). These can then be collected and organised into lines, so that it is easier to read the data. As a development, identical cut out squares/rectangles of paper of different colours can be used with the children taking the correct colour as it relates to them (e.g. choosing the colour for their eyes/hair colour etc.) while also progressing towards using a specific colour for a specific criterion (“Take a blue square if you walked to school today”). Thus, the children should begin to appreciate the need to label the graph, axes etc so that the meaning of the represented data can be correctly interpreted.

HINT: A common confusion among children when making vertical graphs of any type is that the pictures/blocks start at the top and go down; an understandable misconception when you consider that in most other activities we work from the top down! A simple way to show how vertical graphs are formed, is to demonstrate, using a concrete Connect 4 type game, how the first counter in each column falls to the bottom and subsequent counters in that column build up from there. If you don’t have an actual Connect 4 game in your classroom you could use an interactive type such as this one here. 

Abstract: the final stage, where the focus is primarily on numbers and/or digits e.g. identifying how many, how many more prefer this than that etc.

Further Reading and Resources

• Dear Family, your Operation Maths Guide to Data  includes practical suggestions for supporting children, and links to a huge suite of digital resources, organised according to class level.
• The Operation Maths Digital Resources have specific resources designed to support this strand, and have suggestions of online games in the Weblinks section. Full details of these can be found in your TRB. Click here for the Quick Start Guide to the Digital Resources
• Virtual Maths Manipulatives for Data: Lots of tools that can be used in many different ways to explore these concepts.
• Download the PDST Data and Chance Handbook for Teachers
• Early Math Collaborative: Check out the data analysis suggestions in their ideas library
• Teaching Data in Elementary School: A thought-provoking article from the Recovering Traditionalist
• Representing and Interpreting Data Board on Pinterest

Digging Deeper into … Representing and Interpreting Data (3rd-6th)

For practical suggestions for families, and links to useful digital resources, to support children learning about the topic of data, please check out the following post: Dear Family, your Operation Maths Guide to Data

Data Analysis Process

Data analysis, whether at lower primary, upper primary, or even at a more specialised level of statistics, is essentially the same process:

• It starts with a question, that doesn’t have an obvious and/or immediate answer. Information is then collected relevant to the question.
• This collected information or data is represented in a structured way that makes it easier to read.
• This represented data is then examined and compared (interpreted) in such a way as to be able to make statements about what it reveals and, in turn, to possibly answer the initial question (if the question remains unanswered, it may be necessary to re-start the process again, perhaps using different methods).

Thus, every data activity should start with a question, for example:

• What is the most common eye/hair colour in our class?
• Which fruit/pet do we prefer?
• How did we come to school today?
• What candidate would we vote for?
• What is the temperature each day?
• How many children are absent from school every day?

When choosing a question it is worth appreciating that some questions might not lend themselves to rich answers. Take, for example, the first question above; once the data is collected, and represented, there is not that much scope for interpretation of results other than identifying the most common eye/hair colour and comparing the number of children with one colour as being more/less than another colour. However, other questions might lead to richer answers, with more possibilities to collect further information, to make predictions and to create connections with learning in other areas. Take, for example, the question above about travel; the children could be asked to suggest reasons for the results e.g. can they suggest why they think most children walked/came by car on the day in question, whether weather/season/distance from school was a factor and to suggest how the results might be different on another day/time of year. Thus, the children are beginning to appreciate that data analysis has a purpose i.e. to collect, represent and interpret information, so as to answer a question. And, in the case of the questions about temperature and number of absences, the children may begin to appreciate that it is too much to give the specific details for each individual day and that a figure to represent a larger set of numbers (eg the average) is preferable in some situations.

Content overview

A quick glance at the curriculum content for representing and interpreting data for these classes, reveals the following:
3rd class: pictograms, block graphs, bar charts
4th class: pictograms, block graphs, bar charts and bar-line graphs incorporating the scales 1:2, 1:5, 1:10, and 1:100
5th class: pictograms, single and multiple bar charts and simple pie charts; calculating averages
6th class: pie charts and trend graphs; calculating averages

In Operation Maths for 3rd and 4th classes, representing and interpreting data is specifically taught in September, at the beginning of the school year, so that the children are enabled to incorporate these skills into other subject areas where possible e.g. reading and interpreting tables and graphs, collecting and displaying data in science, geography etc.

In 5th and 6th classes, representing and interpreting data is taught later in the school year, after the children have encountered degrees in lines and angles and the circle in 2D shapes, as this content is necessary prerequisite knowledge for creating pie charts. In these classes, representing and interpreting data is also taught as a double chapter (two week block), to allow for the extra time required to explore averages.

CPA

As with every topic in Operation Maths, a CPA approach is also recommended for representing and interpreting data:

Concrete: Using real objects to sort and classify eg the number of different colour crayons in a box, the different type of PE equipment in the hall etc; using unifix cubes, blocks, cuisinere rods etc to represent data; using cubes to introduce and explore the calculation of averages.
Pictorial: using multiple copies of identical images to make pictograms; using identical cut out squares/rectangles to make block graphs etc, using folded circles to make pie charts, using bar models to calculate averages.
Abstract: the final stage, where the focus is primarily on numbers and/or digits eg reading and interpreting the scale on a graph where all the scale intervals are not given; calculating averages without pictorial or concrete supports.

Interpreting data

For children to become comfortable interpreting tables and graphs it is vital that they have plenty of opportunities to look at and read a variety of tables and graphs. This shouldn’t be limited to just the tables and graphs in their maths books. In particular, data sets that are relevant to them, such as soccer league tables can be a great way to encourage the children to appreciate how relevant this strand units is to them.

Utilize every opportunity to expose them to real-life examples of data from print and digital media and use purposeful questions to highlight the features of the graph:

• What is the title of this graph/chart?
• How is the information displayed? Horizontally or vertically?
• What type of chart/graph was used?
• Why do you think this graph type was chosen? What other types would have been suitable?
• What key information is required to interpret the data (eg scale intervals, labels on the axes, a key for piecharts)?
• Is there information missing that would have been useful to get a better insight into the data?

The children can be asked to create questions based on the graph/chart and swap with a partner to answer. When they become adept at producing charts themselves (see next section) they can also be asked to represent the data using a different chart type.

One of the most common mistakes that children make when interpreting graphs is misreading the scale. Always draw children’s attention to this first, and ask them to identify the scale interval and what it means for the bars/blocks/points etc on the graph. The graph quiz on That Quiz provides lots of extra practice for this skill. The quizzes are also very customisable, with options to show pictograms, bar charts, trend graphs (line) and pie charts (circle), easier or normal content, and a variety of question types. Another similar activity is this one from MathsFrame which offers three different levels of questions on bar charts.

A very interesting  and very different way to explore interpreting data is to show the children graphs where much of the key information is missing initially, but is then slowly revealed as the children share their thoughts and ideas. Following on from Brian Bushart’s work on numberless word problems, many teachers have used graphs to create “slow reveal” activities or “notice and wonder graphs”, and have very generously shared these online for other teachers to use. Some of these include:

• What do you notice? What do you wonder? Multiple bar chart
• What do you notice? What do you wonder? Simple bar chart for first grade
• What do you notice? What do you wonder? Bar chart for second grade
• Candy-themed graph from Brian Bushart

Representing data

As mentioned previously, where suitable children should begin to represent data themselves using concrete materials. They can build block graphs using cubes or blocks, laid flat on a piece of paper or their Operation Maths MWBs. These should all start from the same baseline and the children should also write in labels for the axes and a title.

As a development, they can then trace around the stacks of cubes and remove the cubes to have a pictorial representation of the concrete. Using cubes like this to represent 1:1 quantities can in turn lead children to see a need for one cube to represent more than one, ie scales of 1:10, 1:5 etc, especially if there are not enough cubes to represent the data or there is not enough space.

The next step could be to have small squares or rectangles of identical pieces of paper which can then be pasted onto a page to display the information. This can work particularly well for pie charts; cut out a circle of paper and divide it by folding into eighths; the circle can be left whole and the folds outlined in pencil/marker or the eighths can be cut up. A groups of eight children can then use either of these to show data like their favourite ice-cream flavour or TV programme. In this case, because the amount of data gathered is limited, the choices/categories should be limited, also, to three or four.

If the children are also collecting the data to make a graph or chart, they will need to come up with a system to accurately collect and record this data. This will usually involve compiling a type of table with three columns; the first column to list the categories, the second to record tally marks and the third to total the tally marks. When introducing tally systems the children could use lollipop sticks to explore and make tally marks.

For children, drawing their own graphs can present many difficulties. Some common mistakes that can be made include:

• Incorrectly transferring the data from the table to the graph.
• Omitting the graph title and/or category titles on the axes.
• Using an inappropriate scale for a specific graph.
• Not setting the scale at regular, even intervals
• Zero being incorrectly located somewhere other than at the base line/axis.

And in other cases, it can just be a lack of neatness and exactness that reduces the quality, and readability, of a hand-draw graph. To overcome the difficulties associated with hand-draw graphs, the children could use either an online or offline computer application, all of which can produce very impressive results. Listed below are a small sample of those available; click on any of the links to access a tutorial or the application itself.

• How to make a graph using MS Excel: a tutorial
• How to make a graph using Google Docs/sheets: a video tutorial.
• Interactive programme to create line (trend) graphs
• Interactive programme to create bar/pie charts
• Create a Graph: Online graph creation facility that also allows you to print finished product.

Calculating averages

Averages are introduced for the first time in 5th class and the children should have ample opportunities to explore this concept concretely and pictorially, before being given the formula to calculate the average of a set of numbers. Initially, the concept should be introduced as sharing amounts out to be fair/balanced:

Through plenty of concrete and pictorial opportunities to balance these separate quantities, it is hoped that the children begin to see a connection between the total number of items and the balanced quantity or average:

Bar models, one of the key problem-solving strategies used in Operation Maths, are very useful here, where comparison models can be used to compare the total of the averaged quantities with the total of the individual quantities. They are also used in Operation Maths 6 to calculate the extra number(s) when the average increases or decreases, a concept which can be very difficult to reason if no pictorial structures are used to help visualise the relationships.

You can also check out this video to see how bar models can be used to solve averages:

Further Reading and Resources

• Dear Family, your Operation Maths Guide to Data  includes practical suggestions for supporting children, and links to a huge suite of digital resources, organised according to class level.
• The Operation Maths Digital Resources have specific resources designed to support this strand unit. Full details of these can be found in your TRB. Click here for the Quick Start Guide to the Digital Resources.
• Virtual Maths Manipulatives for Data: Lots of tools that can be used in many different ways to explore these concepts.
• PDST Data and Chance Handbook for Teachers
• Data and chance in the world around us: materials from the PDST workshop
• NRICH: selection of problems, articles and games for data
• Teaching Data in Elementary School: A thought-provoking article from the Recovering Traditionalist
• Representing and Interpreting Data Board on Pinterest