COMPARING MEANS: IS YOUR FOOT REALLY THE SAME LENGTH AS YOUR FOREARM?
One of the first statistical measures children encounter within the strand of Data is the (arithmetic)
‘mean’. The mean is commonly called the average in primary classrooms. However, the mean,
median and mode are all types of averages because they all summarize the data in different ways. To
avoid confusion, throughout this article we shall refer to the average as the mean.
The Primary School Mathematics Curriculum (PSMC, p. 109) introduces the mean in 5th class. The
procedure for finding the mean is relatively simple and is commonly referred to as ‘add them all up
and divide by the number of numbers’. Research indicates that the majority of children can compute
the mean without understanding what the mean represents and when you might use it. The only
reference to the mean in the PSMC is to the algorithmic procedure. However, in order for pupils to
make a smooth transition to secondary education particularly in light of the introduction of Project
Mathematics, they need to really understand the concept of mean.
So, what is involved in understanding the mean? Two types of understanding is required, namely
procedural understandings relating to the algorithm (i.e. mathematical properties) and conceptual
understanding of what the mean represents (i.e. statistical properties).
To explore the mathematical properties of the mean, let’s look at the following example:
Six children counted the number of pets they had. John and Anna had 0 pets, Cian had
1, Elsa had 3, Rohan had 4, and Moira had 6. The mean number of pets is 2.33 pets.
Property 1.
The mean is located between the extreme values
Note: The mean of 2.33 falls between the lowest (0 pets) and highest (6 pets) values.
Property 2.
The mean does not necessarily equal one of the values in the data set.
Note: None of the children actually had 2 (or 2.33) pets.
Property 3.
The mean can be a fraction that has no counterpart in physical reality.
Note: The mean of 2.33 pets is not physically possible in reality. Children find this difficult to
understand particularly when it occurs in ‘family size’ data.
Property 4.
The mean is representative of the values that were averaged
Note: Each value in the data set influences the mean. If we change any one value the mean changes.
Property 5.
When calculating the mean, any value of zero must be taken into account
Note: Some children mistakenly ignore zero in their deliberations.
The statistical properties of the mean receive little attention. Research has shown that it is important to
postpone the introduction of the algorithm for finding the mean until children have a conceptual
understanding of the mean. There are different ways to think about the mean. In our lesson we use a
levelling interpretation of the mean. So, when thinking of the pet data above, the mean is the number
of pets that each student has if we were to find the total number of pets (including 0 pets) and divide
them equally between the children. Thinking of the mean using the levelling out analogy facilitates
children to discover and develop the mean algorithm for themselves.
A classroom mathematical exploration of the mean
The remainder of the article, outlines a series of Data explorations, spanning between 3 and 5 lessons,
to develop conceptual understanding of the mean. These explorations were field tested with 5th class
children in Limerick city schools.
It is intended that the sequence of explorations will facilitate children to develop:
Functional understanding of why we use the mean (i.e. to represent a set of data)
Conceptual understanding of the mean – understand where the formula comes from (e.g. levelling
out model of mean)
Procedural/computational knowledge of how to calculate the mean (i.e. the algorithm)
While various real contexts could be used, this article focuses on activities which examine the
accuracy of the assertion that a person’s foot is the same length as one’s forearm.
Introducing the scenario
Present the assertion: The length of the human foot is the same length as the human forearm.
Discuss the assertion. Is it true or false? Why? Have children record their beliefs.
Prior to starting these explorations, we have found it useful to (a) revise the procedure for rounding
decimal numbers, and (b) place children in even groups (e.g 4 or 6 in a group). From our experience,
we found that steps 1-8 take one class period, steps 9-10 a second, and 11-14 a third class period.
Physical determination of the mean to test the assertion
The first lesson focuses on determining the mean value for the length of a foot (in each group)
through the following sequence of activities:
1. Explain that each group will explore whether, for their group, the length of their feet is the same
as the length of their forearms. Present each child with two strips of paper.
2. Have each child measure their foot and cut a strip of paper that matches the length of their foot.
3. In groups, record and discuss the different lengths. What was the shortest/longest foot?
4. Encourage each group to come up with one value that fairly represents the length of a foot. We
call this a representative value. Have each group record their representative value and be prepared
to share and justify their particular value. [We found children tend to use the most frequently
occurring value (i.e. the mode) or a value that falls in the middle of the range of values (i.e. the
midrange)].
5. Explain that they are going to find a representative value for their group by levelling out the
values for their individual foot lengths.
6. In each group have children tape their foot strips end to end (see image 1). Use the long taped
strip and fold it into the number of pieces equal to the number of children in the group (i.e. into
four pieces of equal length if there are four in the group).
7. The length of each equal piece is the mean.
8. Extension/discussion: Children can compare their individual foot length to the mean value and
discuss why there are some feet lengths above and below the mean.
Image 1: Taping the individual paper strips representing the foot length of each group
The next stage, usually the second lesson, focuses on determining the mean length of the forearm in
order to facilitate children to test the assertion that the foot is the same length as the forearm.
9. Repeat steps 1-8 this time focusing on the length of the forearm.
10. Each group now has a mean group value for their feet and their forearms. Using the strips
compare the values (see image 2) and ascertain if the original assertion was true: In their group, is
the mean foot length equal to the mean arm length? Was their original prediction correct?
[You may find that some groups find the means to be similar whereas others will find different means.
Discuss why differences between groups occur (Measurement and rounding errors etc.)]
Image 2: Comparing the paper strips representing the mean foot and forearm length for a group
Introducing the mean algorithm
11. Discuss whether the assertion might be true for the entire class. Is the mean foot length equal to
the mean arm length for the entire class?
12. Ask how we might find the mean foot /forearm length for the entire class. Would the paper strip
method be accurate? Easy to manage? Why? Why not?
At this stage of the discussion in each of the four different classrooms we visited, there was at least
one child who suggested that, rather than use the paper strip method, it would be easier to add up the
[foot/forearm] lengths for everyone in the class and divide by the number of people in the class. If
your class are not reaching this conclusion by themselves (allow a little time!), try the next step to
support the children in linking the mean algorithm to the activities carried out.
13. Tell the children that that you are going to imagine finding the mean for the entire class using the
paper strip method. Imagine making the long paper strip for everyone in the class (for example 28
children). How long would the strip be? How would you find out? The total length would be the
sum of the lengths of each individuals foot/forearm. Imagine dividing the long strip into 28 equal
pieces. To find the length of one section after the strip was divided into 28 equal pieces, divide the
total length by 28.
14. Using calculators, find the mean foot and forearm length for the entire class. Is the assertion true?
Why use this method to explore the mean?
Too often we launch into teaching the mean by introducing the algorithm and never discuss why we
might want to find the mean of a data set. This exploration helps children see the mean as one single
value that can be used to represent a set of numbers. It also identifies the mean as a useful measure to
use if we want to compare data sets. Determining the mean using the paper strip method appeals to all
types of learners as it gives a hands-on and visual way to experience the mean of a data set . This
method is a natural pre-cursor to the algorithm. This approach also provides a context that children
can draw upon when exploring the properties of the mean (in particular properties 1-4).
Enrichment activities to extend learning or provide challenge
• Ask ‘what happens if’ questions. Using the pet data above, ask ‘what happens if … a new
number 0 is added? a new number 23 is added? two new numbers 1 and 2 are added?
• Use technology to explore properties of the mean using a dynamic environment.
o Compare properties of the mean and median using technology
http://www.nctm.org/standards/content.aspx?id=26777
o Use the mean and median applet to investigate the mean for a set of data that you create.
Have individual/pairs/groups explore what happens to the mean when values are added to
a data set. http://illuminations.nctm.org/ActivityDetail.aspx?id=160
• Use children’s literature to provide a problem context for the mean. The story And the Doorbell
Rang by Pat Hutchins can to be to develop the fair share notion of the mean by sharing a number
of cookies equally between people. You can make a pile of cookies and model the story as you
read the book. This method also leads naturally into the development of the mean algorithm.
• Present the children with a mean value. Have them construct a data set that could reflect the
mean. For example: If there are 4 children in a group and their mean height is 135cm, what
could be their four heights? Remind them that there are many answers that are correct.
Authors: Aisling Leavy, Mairéad Hourigan and Áine McMahon lecture in mathematics education at
Mary Immaculate College.
Acknowledgement: Sincere thanks to the co-operating pupils and staff in Scoil Mhathair Dé
(Limerick) and St. Michael’s N.S. (Limerick) and as well to Joanne Bourke, Lisa Kelly, Aisling
Kennedy, Sarah McMahon and Sarah Mulcrony for their contribution to Lesson Study.