Content uploaded by James Russo
Author content
All content in this area was uploaded by James Russo on Mar 02, 2020
Content may be subject to copyright.
3APMC 25(1) 2020
James Russo
MonashUniversity
<james.russo@monash.edu>
Designing and scaffolding
rich mathematical
learning experiences
with challenging tasks
In my capacity running professional learning sessions
in schools I have, on occasion, had the privilege of
being invited to step into another teacher’s classroom,
and lead a mathematics lesson. More often than not,
these teachers are interested in how to teach with
challenging tasks. is is testament to the power of
teaching through problem solving, and the inuence
of Australian teacher-educators and researchers such
as Peter Sullivan, Doug Clarke and Charles Lovitt (and
many others) have had on shaping what is considered
good practice when teaching primary mathematics.
is article describes my experience facilitating
a challenging task in a Year 4 classroom focussed
around exploring relationships between dierent linear
counting patterns. e second part of the article briey
describes how this task could be expanded into a
four-lesson mini-sequence of learning through drawing
on principles of variation theory. However, before
proceeding, it is perhaps necessary to briey describe
the research underpinning challenging tasks and their
pedagogical characteristics.
Pedagogical characteristics of
challenging tasks
Challenging tasks can be viewed as a subset of
problem-solving tasks that possess particular design
characteristics. Summarising the work of Peter Sullivan
and colleagues (e.g., Sullivan et al., 2011, 2015;
Sullivan & Mornane, 2014), at least four distinctive
claims can be made about the design characteristics
of challenging tasks.
1. As students are expected to plan their
own approach to the task and contribute
to a collaborative learning environment,
it is critical that the task has multiple solution
pathways and perhaps multiple solutions,
both to support student agency and to allow
for students to compare and contrast their
approach with peers.
2. As this pedagogical approach is premised
on the idea that students learn best when
provided with opportunities to struggle
and spend time in the ‘zone of confusion’,
all students should experience at least some
important aspects of the task as mathema-
tically challenging.
3. As a single task has been designed to function
as the main focus of the lesson, it needs to
meaningfully engage students in important
mathematical work for a substantial period
of time (e.g., 20 minutes).
4. As all students are expected to participate in
the lesson regardless of their demonstrated
(or perceived) mathematical ability, the task
is generally prepared with accompanying
enabling and extending prompts.
Enabling and extending prompts are a technology
to support dierentiated learning experiences whilst
allowing all students to learn mathematics through
problem-solving (Sullivan, Mousley, & Jorgensen,
2009). Enabling prompts are designed to ensure a
task is accessible to a larger range of learners through
changing how the original problem is represented,
helping the student connect the problem to prior
learning, and/or removing a step in the problem.
It is worth emphasising that following engagement
with the enabling prompt, the general expectation
is that the student will then return to the main task.
Extending prompts, by contrast, are designed for
In response to increasing teacher interest in how to design and implement effective chal-
lenging tasks, James presents the Launch/Explore/Discuss model that builds on the work of
leading Australian educators and researchers Peter Sullivan, Doug Clarke, and Charles Lovitt.
4APMC 25(1) 2020
students who complete the original task. It exposes
these students to an additional task that is more
challenging; however that requires them to use similar
mathematical reasoning, conceptualisations, and
representations (Sullivan et al., 2009).
ere is substantial evidence to support many of
the assumptions underpinning the design characteristics
of challenging tasks. For example, learning appears to
be enhanced when students are provided with oppor-
tunities to construct their own approach to problems
compared with being taught explicit procedures
(Jonsson, Norqvist, Liljekvist, & Lithner, 2014).
Such opportunities lead to larger gains in mathematical
performance even when students do not arrive at a
solution for a given task (Kapur, 2014). e power
of challenging tasks is strongly connected to their
capacity to support episodes of ‘productive struggle’
(Pasquale, 2016).
Productive struggle can be understood as a precon-
dition for facilitating ‘aha’ moments; those moments
where learning almost becomes tangible, and which
tend to be treasured by teachers and students alike. In
the lesson described in the next section, one particular
Year 4 student Damon had an ‘aha’ moment that will
likely stay with me for many years to come. As is
hopefully clear from my elaboration, it was the struc-
ture of the task and my willingness to allow students
to struggle, that, more than other factor, led to the
positive learning experience for Damon and many
of his classmates.
The lesson context
e lesson I am describing took place in a government-
run primary school in regional Victoria, Australia in
a Year 4 class (with 28 students). We had around 50
minutes for the session. e lesson was designed to
follow the launch-explore-discuss (summarise) structure
advocated by proponents of teaching through problem-
solving (Lampert, 2001; see Figure 1).
In consultation with the classroom teacher before-
hand, we had established the learning foci for the lesson.
e primary learning focus was for students to explore
the relationship between dierent linear counting
sequences (i.e., dierent skip-counting patterns). In
particular, students would be expected to distinguish
between numbers that appear in many dierent
sequences (i.e., numbers with many factors) and num-
bers that occur in few sequences (e.g., prime numbers).
I value this learning focus because it makes it clear that
lessons concerned with number patterns are not only
about uency, and that the prociencies
of understanding, reasoning and problem solving can
be considered equally relevant. A secondary learning
focus was for students to be able to recall uently
relevant counting sequences to solve a problem.
Before entering the classroom, I made sure I had all
the materials I needed to support the lesson including
dice (6-sided, 10-sided, 12-sided, 20-sided), number
charts (covering numbers 1-100), as well as enabling
and extending prompts.
Phase 1: Launch
e rst phase of the lesson began with me posing the
Lucky Dice task to students (see textbox), displaying
the task on the electronic whiteboard and reading
it aloud. Students were invited to ask any clarifying
questions. ey were then instructed that they would
work on the task independently, without any support,
for ve minutes. Students were told that they could
have access to a number chart, and a 6-sided dice,
if they thought these would be useful.
Figure 1. Overview of structure of lesson.
• Read problem
• Clarifying
questions
• 0–10 min
independent work
• 5–10 min group
enabling prompt
• 10–30 min independ-
ent/collaborative work
• Five student solutions
discussed; least to
most sophisticated
Discuss (15 min)Explore (30 min)Launch (5 min)
The task: Lucky Dice (Lesson 1)
My dad oered me a deal. I choose any number
on a hundreds chart. He’d then roll a 6-sided
dice, and we’d count by whatever number he
rolled (from zero). If we land on my number,
he’d give me 10 dollars. If we skip my number,
I’d give him 10 dollars. What are some good
numbers I could choose? Should I take the deal?
Russo
Designing and scaffolding rich mathematical learning experiences with challenging tasks
5APMC 25(1) 2020
Phase 2: Explore
I set a countdown timer for ve minutes in a manner
that was visible to students (who were used to this
routine—although not in a mathematics class focussed
on problem solving), and proceeded to roam around
the classroom to gauge how students were approaching
the problem. e purpose of giving them time to work
on the problem independently before collaborating, or
seeking support, was to allow students to gauge their
own understanding of the problem, and to consider
what relevant prior learning they may be able to draw
on. Importantly, this ve minutes is likely to be a
frustrating time for many students. is is intentional.
We are hoping that students will enter what has been
termed the “zone of confusion” (Clarke, Cheeseman,
Roche & Van Der Schans, 2014, p. 58). It is based on
the assumption that getting students to expend eort
“in order to make sense of mathematics, to gure
out something that is not immediately apparent”, is
integral to the learning experience (Hiebert & Grouws,
2007, p. 387). As students become more used to
experiencing mathematics taught in this manner, I
might look to extend this initial ‘independent work
time’ following the launch phase to, for example, 10
minutes (depending on the task at hand, and the age
group of the students).
After ve minutes had elapsed, I instructed students
that anyone struggling to make progress could join
me on the oor where I would share with them the
e group gathered on the oor in front of an inter-
active number chart (Splat Square) on our electronic
whiteboard. I chose the number 13, marked it on the
chart, and students who had joined me took turns
rolling the dice. We used the interactive number chart
to keep track of our counting patterns, and I recorded
our results. We ‘won’ three times (we rolled three ones)
and lost seven times, meaning that we had to pay dad
$40. I left the students with the prompt ‘Can you
choose a better number than 13?’.
rst ‘hint sheet’ (the rst enabling prompt). Ideally,
students access enabling prompts of their own volition
independently of the teacher (Russo, 2018), however
on this occasion it was appropriate to introduce the
prompt to a small group of students to clarify any
ongoing confusion regarding the actual mechanics of
the task. is was particularly the case because this
cohort of students were not used to tasks being posed
in this manner, and were accustomed to more teacher
explanation than I had provided. Around ten or so
students joined me on the oor, with a few others
listening from their tables. I read out the rst enabling
prompt (see textbox).
Figure 2. Enabling prompt 2.
Enabling prompt 1
What if I chose the number 13? Roll the dice 10
times, and see how many times I win the bet.
Can you choose a better number than 13?
What counting
patterns can
you see in
these charts?
Russo
6APMC 25(1) 2020
e rst enabling prompt had taken me around ve
minutes to administer in the small group setting,
meaning that students had now spent 10 minutes in
the explore phase of the lesson. I stopped the class and
let them know that a second ‘hint sheet’ (Enabling
Prompt 2) was available at the front of the classroom
should they desire it (see Figure 2). I also let students
know that at this stage, they could choose to work
collaboratively with other students in groups of two
or three. A majority of students began to work in loose
groups, with several students moving between under-
taking their independent investigations, and sharing
their thinking and ndings with a partner.
It is worth noting that Enabling Prompt 2 is
essentially sacricing our secondary learning objective
(i.e., for students to be able to recall uently relevant
counting sequences to solve a problem) in order
to allow students to focus on the primary learning
objective (i.e., for students to explore the relation-
ship between dierent linear counting sequences).
Prompts have the power to direct and shape the
learning experience, and should serve to focus student
attention toward the key idea inherent in the lesson.
Consequently, it is vital that a teacher has given at least
some thought to a potential hierarchy of intended
learning objectives when choosing or developing
prompts during the lesson-planning phase (Russo &
Hopkins, 2017). During the remainder of the explore
phase, around half a dozen or so students accessed this
second enabling prompt.
e classroom teacher and I continued to roam the
classroom. It was around this stage that many students
had begun to identify numbers that allowed them to
win the bet 5 out of 6 times (numbers such as 12,
24 and 30). In some instances, students had ceased
working on the problem, satised with having found
a number that gave them a ‘good chance’ of winning
the bet. ese students were given the additional
provocation: “Can you prove that there is a number
on the chart that will guarantee me winning the deal
every single time?”.
Around 15 to 20 minutes into the explore phase of
the lesson, several students had determined that choos-
ing the number 60 guaranteed you winning the bet.
As one student stated: “60 is special because it belongs
to all of the counting patterns”. After establishing that
students could justify why this was in fact the case,
and represent their thinking appropriately, they were
provided with the extending prompt (see textbox).
After the explore phase of the lesson had been
running for almost 30 minutes, I brought this phase
Extending prompt
What about if dad rolled a 10-sided dice instead,
but let me choose any number up to 1000? How
would this change the number I might choose?
Am I still guaranteed of winning the deal?
And for an extra challenge: Assuming dad rolled
a 10-sided dice, what is the smallest number I
can choose that guarantees me winning the deal?
to the end. Students were invited to bring their work
to the oor to discuss their solutions.
Phase 3: Summarise and discuss
During the explore phase, one of the responsibilities of
the teacher whilst monitoring student work on the task
is to select several students to present their work to
the class, and think about how these student responses
should be sequenced. e teacher also has to consider
the mathematical connections both between dierent
student solutions to the task, and the underlying
concepts that are the focus of the lesson. is process
is supported by teachers anticipating potential student
responses ahead of time, which is in turn supported
by them working through the task themselves in their
planning teams. Collectively, these notions of antici-
pating, monitoring, selecting, sequencing and connect-
ing are referred to as ve practices for orchestrating
productive mathematical discussions (Stein, Engle,
Smith, & Hughes, 2008). Although there are many
ways to sequence student work meaningfully, one rule
of thumb might be to get students to discuss their
work samples in order of least to most mathematically
sophisticated, with the teacher making connections
between dierent work samples.
Whilst roaming the classroom, I had selected ve
students to present their work on the Lucy Dice task.
Note Andrew and Jill were asked to present their earli-
er solutions to the task rather than their nal solutions
(which were more heavily inuenced by discussions
with peers), in order to better capture the full range
of mathematical work undertaken by the class across
the session.
Andrew
After participating in the oor group exploring
Enabling Prompt 1, Andrew went back to his table,
and chose the number 14. As he reasoned to the class,
14 was likely to be a better number than 13 because:
Designing and scaffolding rich mathematical learning experiences with challenging tasks
7APMC 25(1) 2020
“All even numbers are pretty lucky, because if we roll
a 2 we land on an even number, and if we roll a 1,
we land on an even number. For odd numbers, we’d
have to roll a 1”. Andrew had proceeded to circle all
of the even numbers on his chart, and declared these
lucky numbers. During our discussion, Peta challenged
Andrew: “But what about 9? We can land on 9 if we
roll a 3?”.
I asked the class if they could think of any other
examples of odd numbers that might be “lucky”? Tony
oered 15, which was a number he himself had consid-
ered “because 5 times 3 is 15”; however, he then said
that he thought 12 was “luckier” than 15. I closed this
part of the discussion by asking the group a question
simultaneously intended to partially validate Andrew’s
thinking, and to consolidate our discussion so far:
“Do we agree with Andrew that even numbers tend to
belong to more counting patterns than odd numbers?”.
e class concurred, and Jill (the next student I had
nominated to share their thinking) was asked whether
she had also decided on an even number.
Jill
Jill chose the number 100 as her nal number; however,
on her chart, she had marked all of the multiples of ten.
She oered: “All numbers ending with zero are lucky
– because they are part of the counting by 5s pattern,
and the counting by 2s pattern. ey’re good numbers
to choose”. In a similar manner to Andrew, Jill had put
forward an explanation to justify the utility of a set of
numbers (rather than one specic number), which is
one of the reasons I asked her to contribute her work.
However, Jill had gone beyond Andrew’s thinking by
relating her explanation more directly and compre-
hensively to counting patterns (rather than even and
odd numbers); although it was clear that she had not
thought much about the fact that some numbers end-
ing in zero belong to more counting patterns than oth-
ers. I was conscious that many students had also chosen
multiples of 10 as their number (in particular 30 and
60), and had employed more sophisticated reasoning
than Jill. Rather than let the discussion evolve to this
next level, I wanted to rst engage Damon’s thinking. I
cut the discussion of Jill’s work sample short, and posed
the question: “Jill thought that tens numbers were good
to choose. I wonder if anyone decided that a number
less than 10 could help us win this bet?”.
Damon
Damon was a student described by his teacher as some-
one who found mathematics very dicult and who was
generally disengaged in mathematics lessons. However,
after participating in the small group that explored
Enabling Prompt 1, Damon had returned to his table
with his dice, determined to nd a better number than
13. Damon had proceeded to play against his dad for
much of the next 20 minutes of the session, simulating
the bet by rolling his dice. He had explored a couple
of dierent numbers before settling on the number 6.
In his words: “6 is special because it is means I have
4 ways of winning the bet – 1,2,3 and 6”. Damon
had ‘won’ $230 after choosing the number 6, and was
visibly excited by the fact that he was ‘beating his dad’.
Despite having diculty with many basic skip-count-
ing sequences (e.g, counting by 3s uently beyond 12),
Damon had still managed to engage in the primary
learning focus of the lesson through accessing enabling
prompts. e next question I put to the class was:
“Did anyone come up with an even better number
than 6?”.
Jennifer
Jennifer was the next student to share her thinking.
Like several students, Jennifer had approached the task
methodically, using her hundreds chart to systemati-
cally count by 1s, 2s, 3s, 4s, 5s and 6s, recording each
counting sequence in a dierent colour. She indicated
that “60 is the only number I landed on 6 times”,
and several students concurred that 60 was indeed the
number that guaranteed me winning the bet. e nal
student asked to share his thinking was Max.
Max
Max indicated that 60 guaranteed me winning because
“1, 2, 3, 4, 5 and 6 are all factors of 60. ere are no
other numbers under 100 that have all these numbers
as factors”. I pressed Max on his thinking – what did
it mean for these numbers to be a factor of 60? Max
replied that it meant we could divide 60 by any of
these numbers and get a whole number answer. Max
had made signicant progress with the extending
prompt. In particular, he had identied 720 as a “very
good number” because it had the numbers 1 to 6 (and
10) as factors, and also 8 and 9. He had identied this
number using the sophisticated strategy of setting out
all the multiples of 60 (60, 120, 180 etc), and using
what he knew about multiplication patterns to identify
which of these numbers were potentially multiples
of 7, 8 and 9. However, because Max’s thinking was
signicantly more sophisticated than most of his class-
mates, it was decided to end the discussion without
considering the extending prompt.
8APMC 25(1) 2020
Consolidating and extending the learning
Recently, advocates of teaching through problem-solv-
ing have emphasised the importance of providing stu-
dents with follow-up experiences to consolidate their
understanding and promote generalising (Sullivan et
al., 2015). ese experiences may take many forms,
including a follow-up challenging task that is of similar
level of cognitive demand (Sullivan et al., 2015),
students undertaking more routine tasks (Hopkins
& Russo, 2017), or a purposeful mathematical game
(as outlined here). Sometimes these consolidating
experiences will occur in the same lesson, other times
they are planned for subsequent lessons. To follow-up
the Lucky Dice investigation, I recommended that
the classroom teacher build a second session around
the game Skip-Counting Bingo (see textbox), which
explores the same concept through a similar context.
Many note that when learning through problem-
solving based approaches, presenting and working
through isolated tasks can be problematic (Anghileri,
2006), and that such tasks need to be purposefully
sequenced to best support student learning (Sullivan
et al., in press).
One design principle for supporting the sequencing
of tasks is variation theory. e premise of variation
theory is that deliberately varying only one aspect of
a task can eectively support learning, as it focuses
student attention on what is dierent between the two
versions of a task (Kullberg, Kempe, & Marton, 2017).
One means of interpreting this idea is to keep the
learning context the same, while varying the concept.
e argument is that this approach can deepen under-
standing and help build connections between dierent
mathematical domains (Sullivan et al., in press).
In our case, we may wish to introduce our Year 4
students to the notion of non-linear counting pat-
terns, specically repeated doubles patterns, through
designing a follow-up task labelled Lucky Dice Again
(see textbox). is extending task is identical in its
presentation to the original Lucky Dice task, except
the mathematical process is now repeatedly doubling
numbers rather than skip-counting. Students exploring
the Lucky Dice Again task will learn that doubling
patterns (i.e., multiplying successive terms in a number
sequence by two) cannot be directly equated with any
particular skip-counting pattern. ey may also notice
that repeated doubling patterns grow much faster than
skip-counting patterns. ese contrasting experiences
are laying the foundation for students to distinguish
between arithmetic and geometric sequences.
Consolidating activity: Skip-counting bingo
(Lesson 2)
Materials: 100-chart, Dice (6-sided, 10-sided,
12-sided, or 20-sided dice, depending on desired
level of challenge)
is game is for 2 to 5 players.
To begin, children each choose ve Bingo
numbers in turn, and mark these numbers on
their 100-chart. Note that players must choose
numbers greater than 10 (or 20).
One of the children rolls the dice. Together,
children begin counting by whatever the number
rolled, using the 100-chart to keep track. For
example, if they roll a four, they would begin
counting by 4’s from zero: 4, 8, 12, 16, 20 etc.
Children stop counting when they encounter a
bingo number. In the game shown in Figure 2,
children would stop counting at 28 if a four was
rolled, as this is the rst Bingo number encoun-
tered. is number is removed from the board.
e dice is rolled again, and a new counting
sequence is explored. For example, if a 3 is
rolled, the group would be counting by 3’s
from zero. ey would again stop when they
encountered a bingo number (42 in Figure 3).
Play continues until one of the players removes
all their numbers and shouts ‘Bingo!’.
Having undertaken this pair of lessons, a reasonable
question might be: “Where should students go next in
their learning?”
Figure 3. Three children begin a game of Skip-Counting Bingo.
Russo
Designing and scaffolding rich mathematical learning experiences with challenging tasks
9APMC 25(1) 2020
The task: Lucky Dice Again (Lesson 3)
My dad oered me another deal. I choose any
number on a hundreds chart. He’d then roll a
6-sided dice, and we’d keep doubling whatever
number he rolled. If we land on my number, he’d
give me 10 dollars. If we skip my number, I’d give
him 10 dollars. What are some good numbers I
could choose? Should I take the deal?
Consolidating activity: Doubles Bingo
(Lesson 4)
Materials: 100-chart, Dice (6-sided, 10-sided,
12-sided, or 20-sided dice, depending on desired
level of challenge)
is game is for 2 to 5 players.
To begin, children each choose ve Bingo
numbers in turn, and mark these numbers on
their 100-chart. Note that players must choose
numbers greater than 20.
One of the children rolls the dice. Together,
children begin doubling whatever the number
rolled. For example, if they roll a four, they
would begin doubling 8, 16, 32, 64, 128.
Children stop doubling when they encounter
a bingo number and this number is removed
from the board.
e dice is rolled again, and a new counting
sequence is explored. For example, if a 3 is
rolled, the group would be doubling from 3
(6, 12, 24, 48, 96, 192). ey would again stop
if/when they encountered a bingo number.
Play continues until one of the players removes
all their numbers and shouts ‘Bingo!’.
deliberately designed to have aspects in common.
For example, the Doubles Bingo game in the fourth
lesson is exploring the same concept as the Lucky
Dice Again task (Lesson 3), in the same context as
the Skip-Counting Bingo task (Lesson 2).
Finally, having explored the Lucky Dice Again task,
there is a need to consolidate students understanding
of repeated doubling patterns. is consolidation
might occur through another game-based activity,
designed as a carefully constructed variant on Skip-
Counting Bingo; Doubles Bingo (see textbox). Again,
the mechanics of the game are identical, but for the
fact that, rather than skip-counting, students are
repeatedly doubling numbers.
Concluding remarks
Teaching with challenging tasks has enormous
potential to support student learning in mathemat-
ics (Sullivan et al., in press). In this article, I have
described my experience teaching with one of my
favourite challenging tasks (Lucky Dice), which, in
my experience, can be used eectively with students
as young as Year 2 right through to Year 6. In the
second part of the article, I have spent some time
outlining how we might build a mini-sequence of
learning from the Lucky Dice task through drawing
on principles of variation theory. is mini-sequence
is summarised in Figure 4, which uses the tasks
discussed in this article as an example of a Challenge-
Consolidate-Extend-Consolidate lesson sequence
structure. Hopefully primary teachers nd this
framing to be of some value when considering how
to plan for multiple connected learning experiences
based around a core challenging task.
Acknowledgement
anks to Larissa Raymond (EdPartnerships
International), for her insightful comments that
supported the revision of this article.
Figure 4. Sequence of four lessons adopting a Challenge-
Consolidate-Extend-Consolidate structure.
Lucky Dice
(L1 Challenge)
Lucky Dice Again
(L3 Extend)
Skip Counting
Bingo
(L2 Consolidate)
Doubles Bingo
(L4 Consolidate)
is sequence of four lessons is represented by the
Venn Diagram displayed at Figure 4. e overlap in
the circles is used to denote when tasks have been
10 APMC 25(1) 2020
Russo
References
Anghileri, J. (2006). Scaolding practices that enhance mathematics
learning. Journal of Mathematics Teacher Education, 9(1), 33–52.
Australian Curriculum Assessment and Reporting Authority (ACARA).
(2015). e Australian Curriculum: Mathematics.
Clarke, D., Cheeseman, J., Roche, A., & Van Der Schans, S.
(2014). Teaching Strategies for Building Student Persistence on
Challenging Tasks: Insights Emerging from Two Approaches to
Teacher Professional Learning. Mathematics Teacher Education and
Development, 16(2), 46–70.
Hiebert, J., & Grouws, D. A. (2007). e eects of classroom
mathematics teaching on students’ learning. Second handbook of
research on mathematics teaching and learning, 1, 371–404.
Hopkins, S., & Russo, J. (2017). Using challenging and consolidating
tasks to improve mathematical uency. In R.Seah, M.Horne,
J.Ocean, & C.Orellana (Ed.), Proceedings of the 54th Annual
Conference of the Mathematics Association of Victoria (pp. 85–91).
Melbourne, Australia: MAV.
Jonsson, B., Norqvist, M., Liljekvist, Y., & Lithner, J. (2014).
Learning mathematics through algorithmic and creative reasoning.
e Journal of Mathematical Behavior, 36, 20–32.
Kapur, M. (2014). Productive failure in learning math. Cognitive science,
38(5), 1008–1022.
Kullberg, A., Kempe, U. R., & Marton, F. (2017). What is made possible
to learn when using the variation theory of learning in teaching
mathematics?. ZDM, 49(4), 559–569.
Lampert, M. (2001). Teaching problems and the problems of teaching.
New Haven: Yale University Press.
Pasquale, M. (2016). Productive struggle in mathematics. Interactive
STEM research+ practice brief. Education Development Center, Inc.
Russo, J. & Hopkins, S. (2017). Class challenging tasks: Using cognitive
Load theory to inform the design of challenging mathematical tasks.
Australian Primary Mathematics Classroom, 22(1), 21–27.
Russo, J. (2018). e challenges of teaching with challenging tasks:
Developing prompts. In G. FitzSimons (Ed.), Proceedings of the 55th
Annual Conference of the Mathematics Association of Victoria (pp.
91–96). Melbourne, Australia: MAV.
Splat Square. https://www.primarygames.co.uk/pg2/splat/splatsq100.
html
Stein, M. K., Engle, R. A., Smith, M. S., & Hughes, E. K. (2008).
Orchestrating productive mathematical discussions: Five practices
for helping teachers move beyond show and tell. Mathematical
inking and Learning, 10(4), 313–340.
Sullivan, P., Askew, M., Cheeseman, J., Clarke, D., Mornane,
A., Roche, A., & Walker, N. (2015). Supporting teachers in
structuring mathematics lessons involving challenging tasks.
Journal of Mathematics Teacher Education, 18(2), 123–140.
Sullivan, P.., Bobis, J., Downton, A., Hughes, S., Livy, S., McCormick,
M., & Russo, J. (in press). Ways that relentless consistency and task
variation contribute to teacher and student mathematics learning. For
the Learning of Mathematics.
Sullivan, P., Cheeseman, J., Michels, D., Mornane, A., Clarke, D.M.,
Roche, A., & Middleton, J. (2011). Challenging mathematics tasks:
What they are and how to use them. Paper presented at the Maths
is Multidimensional (Mathematical Association of Victoria),
Melbourne, Australia: MAV.
Sullivan, P., & Mornane, A. (2014). Exploring teachers’ use of, and
students’ reactions to, challenging mathematics tasks. Mathematics
Education Research Journal, 26(2), 193–213.
Sullivan, P., Mousley, J., & Jorgensen, R. (2009). Tasks and pedagogies
that facilitate mathematical problem solving. In B. Kaur (Ed.),
Mathematical problem solving (pp. 17–42). Singapore / USA /
UK, World Scientic Publishing: Association of Mathematics
Educators.
