Encouraging students to persist when working on challenging tasks: some insights from teachers.
Clarke, Doug ; Roche, Anne ; Cheeseman, Jill 等
The Encouraging Persistence Maintaining Challenge (EMPC) Project
has been working with secondary teachers in Melbourne as they seek to
build persistence in students, during work on challenging mathematics
tasks. We have developed sequences of tasks in various topic areas for
Years 7 and 8, which require students to connect different aspects of
mathematics, to devise solution strategies for themselves and to explore
more than one pathway to solutions. There is an expectation that
students record the steps in their solutions, and justify their thinking
to the teacher and other students. We have surveyed teachers on their
strategies for encouraging student persistence, before and after the
experience of teaching some challenging tasks which we have provided to
them. In this article, we provide examples of the kinds of challenging
tasks our teachers are using with students, and share teacher insights
on helpful strategies.
Introduction
It is important for students to learn mathematics, but currently
too many miss out on the opportunities that successful learning creates
(Kilpatrick, Swafford & Findell, 2001; Thomson, Hillman &
Wernert (2012). While it is possible for everyone to learn mathematics,
it takes concentration and effort over an extended period of time to
build the connections between topics, to understand the coherence of
mathematical ideas, and to be able to transfer learning to practical
contexts and new topics (Sullivan, 2011). We use the term persistence to
describe the category of student actions that include concentrating,
applying themselves, believing that they can succeed, and making an
effort to learn, and we term the tasks that are likely to foster such
actions challenging, in that they allow the possibility of sustained
thinking, decision making, and some risk taking by the students.
The notion of persistence is encapsulated in widely used principles
of effective teaching that recommend that teachers communicate high
expectations to students, which involves posing challenging tasks, and
adopting associated pedagogies such as encouraging students to take
risks in their learning, to justify their thinking, to make decisions,
and to work with other students (Stein, Smith, Henningsen & Silver,
2009; Sullivan, 2011).
The authors of the TIMSS video study of Year 8 classrooms noted
that "Australian students would benefit from more exposure to less
repetitive, higher-level problems, more discussion of alternative
solutions, and more opportunity to explain their thinking." They
noted that "there is an overemphasis on 'correct' use of
the 'correct' procedure to obtain 'the' correct
answer. Opportunities for students to appreciate connections between
mathematical ideas and to understand the mathematics behind the problems
they are working on are rare." They noted "a syndrome of
shallow teaching, where students are asked to follow procedures without
reasons" (Hollingsworth, Lokan & McCrae, 2003, p. xxi).
Yet two concurrent projects with which we have been involved in
recent years found that, on one hand, teachers seemed reluctant to pose
challenging tasks to students and, on the other hand, students seemed to
resist engaging with those tasks, and exerted both passive and active
pressure on teachers to over-explain tasks or to pose simpler ones
(Sullivan, Clarke & Clarke, 2013).
Challenging tasks are important for all students. Pogrow (1988)
warned that by protecting the self-image of under-achieving students
through giving them only "simple, dull material" (p. 84),
teachers actually prevent them from developing self-confidence. He
maintained that it is only through success on complex tasks that are
valued by the students and teachers that such students can achieve
confidence in their abilities. There will be an inevitable period of
struggling while the students begin to grapple with problems but Pogrow
asserted that this 'controlled floundering' is essential for
students to begin to think at higher levels.
Our project
The Encouraging Persistence Maintaining Challenge project (1)
(EPMC) is researching a range of issues, including the kinds of teacher
practice which might encourage students to persist when working on
challenging tasks in mathematics. The EPMC is a collaborative project
involving university researchers and classroom teachers in Victorian
schools. Although the project has involved teachers and students across
Years 4 to 8, only the secondary part of the project will be discussed
here. Further information on the primary aspects of the project can be
found in Roche, Clarke, Sullivan, and Cheeseman (2013).
The timing of this project is important in light of the current
implementation of the Australian Curriculum: Mathematics (AC:M, 2013),
and, in particular, with the focus on the reasoning proficiency, one of
four proficiencies, the others being understanding, fluency, and problem
solving. We believe that the challenging tasks discussed in this article
lend themselves to students developing and using reasoning in a variety
of ways.
The major research question of relevance to this paper is: What
strategies on the part of the teacher (during planning and teaching) can
support students to persist when working on challenging tasks?
Fifty-seven primary and secondary teachers met with the project
team in February 2013, typically with four or five teachers coming from
each school. An overview of the project was given, and teachers were
provided with ten challenging tasks, in the form of detailed lesson
notes. For the secondary teachers, the focus was on tasks involving the
content areas of perimeter, area and volume, at Years 7 and 8. All
lessons were written using the structure shown in Appendix 1 for the
lesson, One Hectare Park. Feedback from teachers to date indicates that
the sections in this structure appear to be helpful, and that the lesson
notes are about the right length.
At different points in the project, we have collected information
from teachers on their experiences. Prior to and after teaching the
tasks, we sought teacher perceptions on strategies to encourage
persistence on challenging tasks.
Examples of our challenging tasks
Each lesson has what we have come to call a main task, and is often
accompanied by an introductory task and consolidating tasks. An
important feature of the documentation is the inclusion of enabling
prompts (for students who have difficulty making a start on the main
task) and extending prompts (for students who find the main task quite
straightforward).
To give a further sense of the kinds of tasks in these lessons, we
include the main task from two other lessons:
* The volume of a rectangular prism is 600 [cm.sup.3]. What might
be the surface area?
* Some websites say that 1 mm of rain on 1 [m.sup.2] of roof is 1 L
of water. The roof of a building is 12 m long and 6 m wide. Assume that
60 mm of rain falls each month. Design a tank, in the shape of a
rectangular prism (like this one), that could hold that amount of rain
water for one month. 1 L is the same as 10 cm x 10 cm x 10 cm. 1000 L is
1 [m.sup.3].
[ILLUSTRATION OMITTED]
[ILLUSTRATION OMITTED]
Following the first two day meeting in February, teachers were
asked to teach as many of these ten tasks as possible, before returning
to share their experiences and student work samples with the larger
group in June. Teachers were discouraged from telling the students how
to solve the problems, and asked to ensure that students had plenty of
time to work on the tasks.
Insights from secondary teachers
In February, before any professional learning input from the
research team and the opportunity to trial challenging tasks, teachers
were asked to respond to a question, framed as follows:
Sometimes when students struggle with a mathematics task, they
choose not to persist. What kinds of things do you believe a
teacher could do in the planning stage of a lesson and during the
lesson that would help those students to persist? Please record as
many as you can.
In the planning stage, teachers could ...
During the lesson, teachers could ...
Teachers were given seven lines for each stem, with a verbal
encouragement to put one thought on each line, for as many of the lines
as they wished to complete. Twenty secondary teachers responded with 94
suggestions for the planning stage and 88 suggestions for during the
lesson, an average of 4.7 and 4.4 respectively per teacher. These were
grouped into categories by two members of the research team. In Tables 1
and 2, the most frequently occurring categories are listed, with sample
comments to elaborate the kinds of responses for each category, for the
planning stage, and during the lesson, respectively.
It is clear that the categories are not necessarily mutually
exclusive, and that distinguishing the 'planning stage' from
the 'during the lesson stage' was not clear cut for some
teachers. The major differences in comments between the two stages were
the emphasis on careful choice of tasks in the planning stage, and
discussion and questioning during the lesson to support students with
challenging tasks.
Secondary teachers' insights after teaching up to ten tasks
In June, following the chance to try out up to 10 challenging
tasks, two different prompts were given, as follows:
In this project, you have trialled a number of challenging
mathematics tasks and encouraged students to persist when working on
them. We are interested in what you believe is the most important change
in your practice that contributes to students persisting, both in the
planning stage and during the lesson.
1. In terms of your planning: please describe one aspect of your
planning for these lessons that is different from the way you planned
previously, and which you believe has helped some students to persist.
2. In terms of your teaching: please describe one aspect of your
teaching behaviour during the lessons, that is different from the way
you taught previously, and which you believe has helped some students to
persist.
The teachers were given a verbal encouragement to provide only one
thought, that is, their most important change in practice that was
different from the way they planned and taught previously. Fifteen
secondary teachers responded, each providing one comment, although some
of these comments were coded into two or more categories, when two
somewhat distinct thoughts were contained in the one statement. Once
again, these were grouped into categories by two members of the research
team.
In Tables 3 and 4, the most frequently occurring category is
listed, with sample comments to elaborate the kinds of responses for
this category, for the planning stage and during the lesson,
respectively. Only two categories are mentioned in each case, because
the request for just one response led to the number of responses being
small.
Discussion
In the initial survey in February, some teachers mentioned the use
of enabling and extending prompts, but by June, after the experience of
using the lessons which contained them, not surprisingly, there was a
greater emphasis on this strategy. Two teachers mentioned that solving
the tasks prior to teaching the lesson was the most important change to
their practice in the planning stage. However, no teacher mentioned this
in February.
In the February survey, no teacher made a comment that they should
hold back from telling or hold back from explicitly teaching, but this
was implied in a number of the comments in June. Also, only one teacher
in February commented that giving students time to think was helpful. In
the June survey however, 7 out of the 15 teachers suggested the most
important change in their practice during the lesson was doing one of
these two things.
Although not among the most frequent comments, during focus group
discussions in June, several teachers mentioned the importance of
clarifying new terms to students. In the case of the One Hectare Park
lesson (see Appendix), for example, teachers indicated that this was an
example of one of the times when clarification or the provision of a
definition is a requirement for students to genuinely engage with the
mathematics. Two such terms were 'hectare' and 'internal
angles'. Jackson, Garrison, Wilson, Gibbons and Shahan (2013)
emphasised the importance during the 'setup' of lessons of
developing a common language, as this was directly related to the
opportunities for students to learn during concluding whole class
discussions.
Teachers also indicated that they sought to introduce a more
personal context to some tasks. One teacher used Google maps to
investigate the question: "How big is our school in area?" as
a means of clarifying the unit hectare. In a survey which teachers
completed on each of the lessons they tried, the One Hectare Park lesson
was rated in the top three lessons for student engagement by all seven
teachers who taught it, and rated in the top three for student learning
by five of those seven teachers.
At the February professional learning days, the term "zone of
confusion" was introduced as something which some teachers might
find helpful in discussions with students about the different stages
they might move through as they work on genuinely challenging tasks. It
is clear that this term resonated with both teachers and students in the
project. From responses to other survey items in June, 9 out of 15
teachers claim to now use the term "zone of confusion" and all
of them either agree or strongly agree that it had the desired effect of
assisting students. Fourteen out of 15 claimed that they explained the
benefits of persistence to their students and 8 of the 14 either agreed
or strongly agreed that this assisted students.
As indicated, the greatest change in the kinds of strategies
offered by teachers after the experience of teaching the challenging
tasks appears to be a focus on holding back from telling students how to
solve problems and giving them more time to think about the tasks.
Conclusion
As a project team, we draw upon the insights of primary and
secondary teachers as recorded in questionnaire responses and
audio-taped focus group discussions, and from our own observations in
developing the following list of strategies which appear to support
students to persist when working on challenging tasks:
* some attempt is made to connect the task with students'
experience;
* the ways of working are explained to students, including the type
of thinking in which they are expected to engage and what they might
later report to the class;
* the teacher communicates enthusiasm about the task, including
encouraging the students to persist with it, but holds back from telling
students how to do the task;
* classroom climate encourages risk taking, teachers expect
students to succeed, errors are part of learning, and students can learn
even if they do not complete the task;
* the lesson is structured to ensure that students have adequate
time to work on the challenging task;
* processes and expectations for recording are made clear,
including encouraging students to make appropriate notes along the way;
* the teacher moves around the class, predominantly observing
students at work, selecting students who might report and giving them a
sense of their role, intervening only when necessary to seek
clarification of potential misconceptions, to support students who
cannot proceed, and to challenge those who have completed the task; and
* there is time allowed for lesson review so that students see the
strategies of other students and any summaries from the teacher as
learning opportunities.
For worthwhile learning in mathematics, students need
mathematically appropriate, engaging and cognitively demanding tasks. At
the same time, the decisions which the teacher makes (in planning and
'on the run') can make a considerable difference in how the
task plays out, the level of persistence shown by students, and the
resulting learning, cognitively and affectively. This article has
provided some insights from teachers into the kinds of decisions which
they make, during planning and during teaching, which appear to maximise
the potential of the tasks for worthwhile learning.
Appendix: One hectare park
A park is enclosed by a fence that has exactly 6 internal
right angles.
The total area of the park is 1 hectare.
What might be the perimeter of the park?
(Give two different answers)
Rationale for the lesson
A key insight for students is to see that the length of some of the
sides of rectangular shapes influences the lengths of the other sides.
This lesson also connects a problem related to angles, hectare as a unit
of measurement, and perimeter.
Year level
7-8
Particular pedagogical considerations
There is a variety of shapes that have six internal right angles
(see below). Note that there are possible shapes that do not consist
solely of 90[degrees] and 270[degrees] angles but calculating perimeters
of such shapes is not expected at this stage.
Explain that 1 hectare is the same as 10 000 m (that is, the same
as a square 100 m x 100 m). Have a discussion of a space locally that
has an area of about a hectare.
Note that a common misconception arises when multiplying large
numbers (2 thousand times 3 thousand is not 6 thousand).
For the students
The meaning of hectare as a unit of area, internal angles and
finding the perimeter of a shape. For one area you can have many
different perimeters.
Introductory task
A park is enclosed by a fence that has exactly six internal right
angles. What might the park look like?
Enabling prompt
A park is enclosed by a fence that has exactly four internal right
angles. The area of the park is 100 [m.sup.2]. What might be the
perimeter of the park?
Extending prompt
If a park is enclosed by a fence that has exactly seven internal
right angles and the total area is 1 hectare, what might make it hard to
calculate the lengths of the various sides?
Consolidating task(s)
A park is enclosed by a fence that has exactly eight internal right
angles. The area of the park is 2 hectares. What might be the perimeter
of the park?
Some possible student solution strategies
There is a variety of shapes that have 6 internal right angles,
although the process of calculating the lengths of the sides is similar
(so long as the internal angles are either 90[degrees] or 270[degrees]).
Some of the possible shapes that have six internal right angles
are:
[ILLUSTRATION OMITTED]
Most students will convert the hectare to square metres. One
possible solution is to see the T shape made from 5 rectangles, each of
which has an area of 2000 [m.sup.2]. Assume that the dimensions are 50 m
x 40 m, the diagram might look like the following.
[ILLUSTRATION OMITTED]
So the perimeter is 540 m.
References
ACARA (2013). Australian Curriculum: Mathematics. Accessed 14
August 2013 from http://
www.australiancurriculum.edu.au/Mathematics/Curriculum/F-10
Cheeseman, J., Clarke, D., Roche, A. & Wilson, K. (2013).
Teachers' views of the challenging elements of a task. In V.
Steinle, L. Ball & C. Bardini (Eds), Mathematics education:
Yesterday, today and tomorrow (Proceedings of the 36th annual conference
of the Mathematics Education Research Group of Australasia, pp.
154-161). Melbourne: MERGA.
Hollingsworth, H., Lokan, J. & McCrae, B. (2003). Teaching
mathematics in Australia: Results from the TIMSS video study (TIMSS
Australia Monograph No. 5). Camberwell, Vic.: Australian Council for
Educational Research.
Jackson, K., Garrison, A., Wilson, J., Gibbons, L. & Shahan, E.
(2013). Exploring relationships between setting up complex tasks and
opportunities to learn in concluding whole-class discussions in
middle-grades mathematics instruction. Journal for Research in
Mathematics Education, 44(4), 646-682.
Kilpatrick, J., Swafford, J. & Findell, B. (2001). Adding it
up: Helping children learn mathematics. Washington DC: National Academy
Press.
Pogrow, S. (1988). Teaching thinking to at-risk elementary
students. Educational Leadership, 45(7), 79-85.
Roche, A., Clarke, D., Sullivan, P. & Cheeseman, J. (2013).
Strategies for encouraging students to persist on challenging tasks:
Some insights from work in classrooms. Australian Primary Mathematics
Classroom, 18(4), 27-33.
Stein, M. K., Smith, M. S., Henningsen, M. A. & Silver, E. A.
(2009). Implementing standards-based mathematics instruction (2nd ed.).
New York: Teachers College Press and National Council of Teachers of
Mathematics.
Sullivan, P., Clarke, D. M. & Clarke, B. A. (2013). Teaching
with tasks for effective mathematics learning. New York: Springer.
Sullivan, P., Clarke, D., Clarke, D. & Roche, A. (2013).
Teachers' decisions about mathematics tasks when planning lessons.
In V. Steinle, L. Ball & C. Bardini (Eds), Mathematics education:
Yesterday, today and tomorrow (Proceedings of the 36th annual conference
of the Mathematics Education Research Group of Australasia, pp.
626-633). Melbourne: MERGA.
Sullivan, P. (2011). Teaching mathematics: Using research-informed
strategies. Australian Education Review 59. Camberwell, Victoria:
Australian Council for Educational Research.
Thomson, S., Hillman, K. & Wernert, N. (2012). Monitoring
Australian year 8 student achievement internationally: TIMSS 2011.
Melbourne: Australian Council for Educational Research (ACER).
Doug Clarke
Australian Catholic University
<doug.clarke@acu.edu.au>
Anne Roche
Australian Catholic University
<anne.roche@acu.edu.au>
Jill Cheeseman
Monash University
<jill.cheeseman@monash.edu>
Peter Sullivan
Monash University
<peter.sullivan@monash.edu>
(1) The Encouraging Persistence Maintaining Challenge project is
funded through an Australian Research Council Discovery Project
(DP110101027), and is a collaboration between the authors and their
universities.
Table 1. Most common strategies in the planning stage for
encouraging persistence on challenging tasks.
Strategy for the Number of Percentage Sample comments
planning stage comments of all
comments
(n = 94)
Differentiation 18 19.1% * Plan according
to their level
* Have extensions
for those who
get it easily
* Plan for
different
abilities
Nature of tasks 13 13.8% * Use open-ended
tasks so that
each student
can start
* Use games that
require skill
but chance
element means
all can safely
attempt
* Choose tasks
that connect to
students'
lives, culture
Teacher 11 11.7% * Identify
knowledge of students who
students may struggle
* Investigate
data available
to get to know
the students
* Plan tasks that
link to
students' prior
knowledge
Explicit 11 11.7% * Use examples
teaching beforehand in
class that have
similar
approaches to
those that
could be used
for the task
* Brainstorming a
range of ways
of approaching
a problem
before starting
on the task
* Provide
terminology/
definitions
Table 2. Most common strategies during the lesson for encouraging
persistence on challenging tasks.
Strategy for Number of Percentage of Sample comments
during the comments all comments
lesson (n = 88)
Discussion/ 23 26.1% * Allow for individual
questining/ discussion to allow
sharing students to clarify
thoughts
* Have a student
explain their
thinking
* Support students by
asking questions
Explicit 180 20.5% * Redefine the
teaching procedure to
individuals/whole
class if necessary
* Assist with basic
skills
* Provide them with
more explanation to
help their
understanding
Differentiation 13 14.8% * Assist and show
students how to do
a question and sit
with them to give
them support for
the next one(s)
* Try explaining in
another way for
strugglers
* Intervene where
necessary to extend
or enable
Teacher 5 5.6% * Encourage students
enthusiasm/ to persist through
encouragement positive language
* Encourage students
to begin and have a
go
* Embrace and
acknowledge attempts
Table 3. Most common new strategies in the planning stage
for encouraging persistence.
Strategy in the Number of Sample comments
planning stage teachers
out of 15
Differentiation 8 * Planning for
enabling prompts
* Having the enabling
prompt and extending
prompt ready to go
* Planned a whole range
of enabling prompts
particularly physical
manipulatives
Solving the task 2 * I worked through the
tasks completely
before teaching
* I completed each task
before taught--
consciously trying to
do so in multiple ways
to predict student
approaches
Table 4. Most common new strategies during the lesson for
encouraging persistence.
Strategy during Number of Sample comments
the lesson teachers
out of 15
Holding back 5 * I am a facilitator of the
from telling students thinking, more
teaching 'behind' than a
teacher of explicit and
closed activities
* Not giving in and telling
them--instead giving
hints and questioning
students when they are
stuck
* I have changed from
helping the students to
encouraging their
thinking. The students
are then going back to
try and work through
[the task]
Think time 2 * The launch phase--asking
students to work in
silence for the first
five minutes
* Use the 1-5 minute
individual think time
more consistently now
