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# Using picture story books to discover and explore the concept of equivalence

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The notion of equivalence is a very important concept for students and should be developed from a young age. This article demonstrates how students can deepen their relational understanding of the equals sign by exploring inequalities within a dice game based on familiar children’s literature.
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Introduction
Students deepen their relational understanding
of the equals sign through exploring inequalities
in this competitive dice game, built around the
familiar fairy-tale e ree Little Pigs and e Big
abilities by choosing more or less challenging dice
combinations. e two follow-up investigations,
based on the story Who Sank the Boat?, are intend-
ed to consolidate (Investigation 1), and further
extend (Investigation 2), student understanding
of the equivalence concept.
Context
Developing a relational understanding of the equals
as meaning ‘the same as’, rather than simply ‘the
answer’. It is a critical aspect of students’ develop-
ment in thinking mathematically that should be
promoted as soon as students begin encountering
number sentences (Karp, Bush & Dougherty,
2014). Such a relational understanding lays the
foundation for algebraic thinking and promotes
exible representations of numbers (Molina &
Ambrose, 2006).
For example, a relational understanding of
the equals sign supports ‘part-whole thinking’,
an important milestone in a young student’s math-
ematical development which involves the student
transitioning away from relying on counting-based
strategies to using partitioning and compensation
(Young-Loveridge, 2002).
is point is appropriately captured by Willis
(2000), in her description of two Grade 1 students
grappling with the number sentences 4 + 2 and 3
+ 3. Whilst Sam understands that he can use his
ngers to compute 4 + 2 = 6 and 3 + 3 = 6, for him
these facts remain unconnected bits of knowledge.
By contrast, Annie appears to grasp the connection
between them, which suggests the foundations for
an understanding of equivalence; in this instance
that 4 + 2 = 3 + 3 = 6. In her own words:
ey both equal 6 because if you take one
o the four and give it to the two, to make it
three, then it is 3 add 3 or you could take one
o the three and give it to the other three and
make 4 + 2. at’s why both have to be the
same. (Willis, p. 32–33)
Many mathematics educators view the frequent-
ly narrow conception of the role of the equals sign
in primary school classrooms as problematic. For
example, Perso (2005) argues that students are
conditioned to “do something now” or “nd an
answer now” whenever they encounter an equals
sign (p. 214). She contends that this action-
oriented, operational understanding of the symbol
prevents students considering its relational aspect,
which in turn impedes the development of
algebraic thinking. She suggests a range of peda-
gogical approaches for attempting to address this
misconception, including: using balance beams
to visually play with concepts of equivalence,
being exposed to practical worded problems which
encourage the use of compensation strategies,
Using picture story books
to discover and explore the concept of
James Russo
Belgrave South Primary School, Vic.
<mr.james.russo@gmail.com>
The notion of equivalence is a very important concept for students and should be developed from a
young age. This article demonstrates how students can deepen their relational understanding of the
equals sign by exploring inequalities within a dice game based on familiar children’s literature.
equivalence
26 APMC 21 (2) 2016
Using picture story books to discover and explore the concept of equivalence
and using partitioning to encourage students to
explore numerical equivalence in its symbolic form.
Despite its importance, developing this
relational understanding of the equals sign can
be extremely challenging, even when a teacher
spends considerable time exploring the concept
in the classroom (Seo & Ginsburg, 2003). One
possible means of laying the foundation for a deep-
er understanding of equivalence may be to provide
students with opportunities to discover this rela-
tional meaning of the equals sign. is discovery
can be promoted through juxtaposing the concept
of equivalence with the concept of inequality (and
the corresponding inequality signs) early in a stu-
dent’s mathematical development (Russo, 2015).
is article will introduce a competitive dice
game, built around the familiar fairytale, e ree
Little Pigs and the Big Bad Wolf, designed to foster
this discovery process. e article then outlines
two follow-up investigations based around the text
Who Sank e Boat? e rst investigation provides
students with a further opportunity to explore and
consolidate the concept of equivalence using a
dierent representation, specically the balance-
beam image suggested by Perso (2005). e
second investigation further extends the concept
of equivalence into a problem context involving
proportional reasoning.
The game:
Three Little Pigs
versus
Teachers may wish to read a version of the fairytale
prior to the activity in order to engage students
before introducing students to the game.
Table1. Suggested dice.
Although the game is best suited to students aged
from six to nine, older children could still benet
from the activity.
Setup
Students should play the game in pairs. e only
equipment they need are various dice and some
paper and pencil (or a whiteboard and whiteboard
marker). e dice they should select depends on
the age group and current ability level of students.
e rules of the game are set out below:
• Rule1: In pairs, one student plays the pigs
and the other student the wolf.
• Rule2: Dice are rolled, and students calculate
their score for that role. For example, using
the Years 3–4 dice, the player representing
the pigs would sum the three 20-sided dice
together, while the wolf would halve whatev-
er number they rolled on their 10-sided 10s
dice. e player with the higher score records
the number sentence (using the greater-than
or less-than sign), and earns a ‘house’.
• Rule3: First to ve houses wins.
• Rule4: If both players obtain the same score,
they both record the number sentence (using
the equals sign), and both earn a ‘house’.
Teaching tips
It is recommended that rules 1, 2 and 3 be shared
with students prior to them playing game. ese
three rules should be presented unnumbered on
strips of card, and displayed prominently in the
classroom (see Figure 1).
If students have diculty during the game,
the teacher should refer the students to the three
game rules. However, rule 4 is best shared with
Suggested dice (with suggested operations in parentheses)
ree Little Pigs e Big Bad Wolf
Years 1–2 ree 6-sided dice
20-sided dice
(total)
Years 3–4
ree 20-sided dice
10-sided 10’s dice
(half)
27APMC 21 (2) 2016
Russo
Figure 1. Introducing the first three rules.
the students only once the game has commenced
or only after students raise the problem of players
obtaining the same score.
More specically, teachers should instruct
students in using the greater-than or less-than
sign appropriately in the pre-game introduction
when the activity is launched. For younger stu-
dents, consider introducing the application of
the greater-than or less-than sign as the “crocodile
always eating the larger amount”. It is recommend-
ed, however, that teachers do not provide students
with explicit instructions on what to do when the
scores are the same. Ideally, the teacher should
let the need to use the equals sign emerge from
students’ own reasoning, and explore this in
more depth during the post-game discussion
(see Figure 2).
a tie during the pre-game discussion, the teacher
can respond something like “Hmmm I wonder if
that will happen? If it does, let me know and we
will decide what to do”. Obviously, if, during the
launch of the activity with the whole class, both
players obtain the same score, the teacher may
need to bring the discussion of rule 4 forward. e
teacher will need three rounds or so to demonstrate
the game to the students. It is worth noting that a
tie is relatively unlikely to occur. (Using the dice
recommended for older students, in a given round
the probability of a tie is less than 2%.)
Some questions for guiding the post-game
discussion appropriate for the rst (and, depending
on the age of the students, possibly second) time
students play the game include:
• What was the score in your game?
• Did anyone have both players roll the
same Wscore during a round?
• What did you decide to do?
Did the game rules help?
• What new rule do we need to include in
the game when both scores are the same?
Teachers working with older students (i.e., Years
3 and 4) can even get students to briey work on
this additional rule in pairs, record it and then share
it with the class. Rule 4 can then be introduced on
a strip of card, and displayed with the other rules in
the classroom.
Figure 2. Discovering the fourth rule.
Get students to play the game again in subse-
quent sessions using all four rules. e game, even
in this relatively simple format, can be revisited on
several occasions. If you feel that students require
further extension, the same basic game mechanism
can support the use of more sophisticated strategies
and concepts involving mental computation.
For example, try playing with ten little pigs (ten
6-sided dice) vs three big bad wolves (three 20-sided
dice); or, if exploring multiplication, three 6-sided
dice that need to be multiplied together (for the
pigs), vs a 10-sided 10s dice (for the wolf).
Example of a game
e game was played in a Year 3 and 4 composite
class using the appropriate dice as previously
described. Two Year 3 students Cada (pigs) and
Samantha (wolf) began a game together. On the
houses each, Cada rolled a 20, 10 and 15 on her
20-sided dice, and Samantha rolled a 90 on her
28 APMC 21 (2) 2016
ten-sided dice. After Samantha halved the number
on her dice, the students realised that they had the
same score (see Figure 3).
As this was the rst time they had played the
game, a great deal of excitement followed, and Cada
yelled across the room “We got the same score, so
we dont know which way the crocodile sign should
face. What should we do?! What should we do?!”
e teacher asked “What do you normally do when
two sides of a number sentence are the same? What
sign would you use?” Samantha replied elatedly
“e equals sign! ey are the same! We use the
equals sign!” e teacher replied that both students
could record the number sentence, and both earn
one house each. e need to use the equals sign only
arose in around one-third of the games, however
these instances provided a fascinating point of focus
for the post-lesson discussion (Note that playing
with the simpler dice, outlined for Grade 1 and 2
students, will result in the equals sign needing to
be used more frequently).
Figure 3. Example of a game: Cada (Pigs) vs Samantha (Wolf).
Consolidating and extending the
concept of equivalence:
Who Sank
The Boat?
Context
Read the classic childrens story Who Sank e Boat?
by Pamela Allen to the class, as a precursor to
launching the following investigations.
e rst investigation, “How can we balance
the boat?”, is designed to consolidate students’
understanding of equivalence. e investigation
explicitly incorporates Persos (2005) suggested
balance beam representation of equivalence and
allows students to tangibly and visually explore the
concept. e open-ended nature of the rst inves-
tigation, and its inclusion of an enabling prompt,
supports dierentiation and ensures it is a poten-
tially suitable activity for students in Years 1 to 4
(Sullivan, Mousley, & Zevenbergen, 2006).
e second investigation, “How heavy is the
mouse?”, is designed to build on the rst investiga-
tion (hence students should have already undertak-
en the rst investigation during a prior lesson).
It is considerably more challenging and is suitable
for older students (Years 3 to 5). It is designed
to extend student understanding of equivalence
through requiring students to apply the concept
to explore interrelationships between unknown
quantities. It involves proportional reasoning and
more closely resembles a formal algebraic problem.
Figure 4. Plasticine ‘models’ of the cow,
donkey, pig, sheep and mouse.
Using picture story books to discover and explore the concept of equivalence
29APMC 21 (2) 2016
Russo
Investigation 1: How can we balance
the boat?
Materials
• Paper and tape to create boats
• Playdough or plasticine to model the animals
• Pencils and paper to draw answers
Describing the problem
One of the reasons the boat stayed aoat so long
is because the animals worked out how to balance
their weights across the boat.
Can you nd a way to get all ve animals,
including the mouse, to distribute their weight
across the boat so that the boat is balanced and
stays aoat? Here is some important information
the problem:
• e cow weighs the same as the donkey
(Cow = Donkey).
• e pig weighs the same as the sheep
(Pig = Sheep).
• e cow and the donkey are both heavier
than the pig and the sheep (Cow > Pig, Cow
> Sheep; Donkey > Pig, Donkey > Sheep).
• e pig and the sheep are both heavier than
the mouse (Pig > Mouse; Sheep > Mouse).
• See how many dierent ways you can solve
the problem.
What do the students need to do?
• Create a boat using paper and tape.
• Model the animals using plasticine
or playdough in accordance with the
above information (see Figure 4).
• Use their animal models and paper boat to
explore solutions to the problem (see Figure 5).
• Record their solutions by drawing them
on paper as they discover them.
• Encourage students to work in pairs or groups
of three to tackle the investigation.
Mathematical reasoning and critical thinking
can be supported by declaring that a solution
may only be recorded when all group members
agree that a particular conguration of animals
would balance the boat. If agreement cannot be
reached by the group on a particular congura-
tion, the teacher should consider photographing
it and exploring it further during the whole-
class discussion (it may provide an opportu-
nity to address a misconception or provide
an example where there is genuine ambiguity
about whether the boat would be balanced).
• ere are theoretically innite solutions to
this challenge, some of which are displayed
in Figure 5. However, the key insight into
the problem is realising that the mouse needs
to be exactly in the middle of the boat.
Enabling prompts for students
• If the mouse got onto the boat on his own,
where would he need to stand to balance
the boat?
• What if the mouse was in the middle of the
Figure 5. Some possible solutions to the ‘How can we
balance the boat?’ investigation.
Investigation 2: How heavy is the mouse?
Materials
•Unixblocks
30 APMC 21 (2) 2016
Describing the problem
Of course, the other reason the boat sank is because
the combined weight of the animals was too heavy
for the little row boat. You have been given some
extra information about the animals’ weights:
• e cow and the donkey are both twice as heavy
as the pig and the sheep (Cow = 2 Pigs, Cow =
2 Sheep; Donkey = 2 Pigs, Donkey = 2 Sheep)
• e pig and the sheep are both ve times
heavier than the mouse (Pig = 5 Mouse;
Sheep = 5 Mouse)
You have been told that the rowboat you have
made can hold up to 100 unix blocks before it
sinks, so the combined weight needs to be less
than this.
• Cow + Donkey + Pig + Sheep + Mouse
< 100 unix blocks
What is the maximum weight the mouse can be
(in unix blocks) to keep the boat aoat?
What do the students need to do?
Students can solve the problem however they like,
however the teacher may wish to encourage
students to physically model the problem with
unix blocks.
• e solution to the challenge is that the mouse
can weigh 3 unix blocks (i.e., 3 + 15 + 15
+ 30 + 30 equals 93, which is less than 100).
Although the challenge has only one solution,
the extending prompt is designed to get stu-
dents to generalise the relationships amongst
the variables (i.e., the animal weights), and
apply proportional reasoning. is process
can be viewed as constituting an elementary
form of algebraic reasoning (Perso, 2005).
• Although some concept of proportional rea-
soning is required to engage with the challenge,
students should be encouraged to pursue the
problem through trial and error. Combined with
the enabling prompt, this should provide many
students with a pathway into the problem.
Enabling Prompts
What if the mouse weighed one unix block?
How much would the pig and sheep weigh?
What about the cow and donkey? How much
weight would there be in the boat altogether?
Extending Prompts
Work out the maximum weight the mouse can
be if the boat can hold up to:
• 200 unix blocks
• 300 unix blocks
• 500 unix blocks
• 1000 unix blocks
• 10000 unix blocks
Conclusion
Building a deeper understanding of equivalence,
and, in particular, grasping its relational aspect is
both critical to developing number sense (Karp et
al., 2014) and potentially very challenging (Seo &
Ginsburg, 2003). It is suggested that playing the
ree Little Pigs dice game, and undertaking the
follow-up investigations using the text Who Sank
e Boat? can help students to engage authentically
with this critical concept.
References
Allen, P. (1982). Who sank the boat? Australia: omas
Nelson.
Karp, K. S., Bush, S. B., & Dougherty, B. J. (2014).
13 rules that expire. Teaching Children Mathematics,
21(1), 18–25.
Molina, M., & Ambrose, R. (2006). Fostering relational
thinking while negotiating the meaning of the equal sign.
Teaching Children Mathematics, 13(2), 111–117.
Perso, T. (2005)in M. Coupland, J. Anderson, & T. Spencer
(Eds.) Making mathematics vital: Proceedings of the
twentieth biennial conference of the Australian Association
of Mathematics Teachers (pp. 209-216). Sydney, Australia:
AAMT.
Russo, J. (2015). Surfs up: An outline of an innovative
framework for teaching mental computation to stu-
dents in the early years of schooling. Australian Primary
Mathematics Classroom, 20(2), 34–40.
Seo, K. H., & Ginsburg, H. P. (2003). “You’ve got to
carefully read the math sentence...”: Classroom context
and children’s interpretations of the equals sign. In A. J.
Baroody & A. Dowker (Eds.), e development of arith-
metic concepts and skills: Constructing adaptive expertise
(pp. 161–186). Mahway, New Jersey: Lawrence Erblaum
Associates, Publishers.
Sullivan, P., Mousley, J., & Zevenbergen, R. (2006). Teacher
actions to maximize mathematics learning opportunities
in heterogeneous classrooms. International Journal of
Science and Mathematics Education, 4(1), 117–143.
Willis, S. (2000). Strengthening numeracy: Reducing risk.
Paper presented at the Australian Council for Educational
Research (ACER), Improving numeracy learning: Research
conference 2000: Proceedings. 31–33.
Young-Loveridge, J. (2002). Early childhood numeracy:
Building an understanding of part-whole relationships.
Australian Journal of Early Childhood, 27(4), 36.
Using picture story books to discover and explore the concept of equivalence
31APMC 21 (2) 2016
... Lamberg and Andrews (2011) indicated that using physical models and learning aids in activities, such as counting edible cookies, could help to visualise the connection between stories, real-life experiences and mathematics. Cutler et al. (2003) and Russo (2016) also considered the use of games, as an extension of the reading, to be an effective tool for connecting stories to mathematical concepts. Within these learning activities in classrooms, by observing the engagement level of students during the reading session, gauging their understanding throughout the lesson, and collecting their opinions on the picture books and the learning, teachers can obtain immediate feedback on their lesson planning, facilitating instant recalibration of their teaching. ...
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Who sank the boat? Australia
• P Allen
Allen, P. (1982). Who sank the boat? Australia: Thomas Nelson.
The development of arithmetic concepts and skills: Constructing adaptive expertise
• K H Seo
• H P Ginsburg
Seo, K. H., & Ginsburg, H. P. (2003). "You've got to carefully read the math sentence...": Classroom context and children's interpretations of the equals sign. In A. J. Baroody & A. Dowker (Eds.), The development of arithmetic concepts and skills: Constructing adaptive expertise (pp. 161-186). Mahway, New Jersey: Lawrence Erblaum Associates, Publishers.
Strengthening numeracy: Reducing risk. Paper presented at the Australian Council for Educational Research (ACER), Improving numeracy learning: Research conference
• S Willis
Willis, S. (2000). Strengthening numeracy: Reducing risk. Paper presented at the Australian Council for Educational Research (ACER), Improving numeracy learning: Research conference 2000: Proceedings. 31-33.
Making mathematics vital
• T Perso
Perso, T. (2005)in M. Coupland, J. Anderson, & T. Spencer (Eds.) Making mathematics vital: Proceedings of the twentieth biennial conference of the Australian Association of Mathematics Teachers (pp. 209-216). Sydney, Australia: AAMT.