Teachers holding back from telling: a key to student persistence on challenging tasks.

Roche, Anne ; Clarke, Doug

Anne Roche and Doug Clarke discuss the importance of developing students' persistence in relation to problem solving during the use of challenging tasks. They provide a useful list of strategies that teachers can use to encourage persistence amongst their students.

We report here on our work with primary teachers to increase persistence as students engage with challenging mathematics tasks as part of the Encouraging Persistence Maintaining Challenge project (EPMC). We developed sequences of tasks in various topic areas for Grades 5 and 6, which require students to connect different aspects of mathematics, to devise solution strategies for themselves, and to explore more than one pathway to solutions. We surveyed teachers on their strategies for encouraging student persistence, before and after the experience of teaching some challenging tasks which we had provided to them. In this article, we provide examples of the kinds of challenging tasks project teachers are using with students, and share teacher insights on helpful strategies for encouraging persistence.

Introduction

For many years, the importance of problem-solving in mathematics education has been well recognised. As Thompson, Battista, Mayberry, Yeatts and Zawojewski (2009) noted,

   Good problems challenge students to
   develop and apply strategies, serve as a
   means to introduce new concepts, and
   offer a context for using skills.

   Problem solving is not a specific topic to be
   taught but permeates all mathematics. (p. 2)

In recent years, there has been a greater emphasis in research and curriculum documents on the important role played in problem solving by challenging or cognitively demanding tasks (Stein, Smith, Henningsen & Silver, 2009).

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.

Sullivan, Cheeseman, Michels, Mornane, Clarke, Roche and Middleton (2011) characterised challenging tasks as those which require students to:

* plan their approach, especially sequencing more than one step;

* process multiple pieces of information, with an expectation that they make connections between those pieces, and see concepts in new ways;

* engage with important mathematical ideas;

* choose their own strategies, goals, and level of accessing the task;

* spend time on the task;

* explain their strategies and justify their thinking to the teacher and other students; and

* extend their knowledge and thinking in new ways (p. 34).

We believe that all students should experience challenging tasks, but sometimes teachers are reluctant to pose challenging tasks to students, and this is compounded by students' reluctance to engage with those tasks (Pogrow, 1988; Roche, Clarke, Sullivan & Cheeseman, 2013; Sullivan, Clarke & Clarke, 2013).

Our current project

The Encouraging Persistence Maintaining Challenge project (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 and Tasmanian schools.

Examples of our challenging tasks

Teachers met with the project team for two full days in February 2013, with four or five teachers attending 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 36 primary teachers, the focus was on tasks involving the content areas of multiplication and division at Years 5 and 6. All lessons were written using the structure shown in Appendix 1 for the lesson Missing Number Multiplication. Each lesson has what we have come to call a main task, and this 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; see Sullivan, 2011).

To give a further sense of the kinds of tasks in these lessons, we include the main task from two other lessons:

* from Patterns without Remainders.

Some people came for a sports day. When the people were put into groups of 3, there was one person left over. When they were lined up in rows of 4, there were two people left over. How many people might have come to the sports day?

* from Working with a Calculator with a Broken Button.

For each expression, write two different expressions that show how to use a calculator with the '4' button broken to do the calculations:

234 x 9 314/3

Teachers were asked to teach as many of these 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 teachers

At different points in the project, we have collected information from teachers on their experiences.

Primary teachers' insights prior to teaching the challenging tasks

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. The 36 primary teachers responded with 172 suggestions for the planning stage and 164 suggestions for during the lesson, an average of 4.6 and 4.4 respectively per teacher. These were grouped into categories by the research team. In Tables 1 and 2, the most frequently occurring categories are listed, with illustrative comments to elaborate the kinds of responses for each category, for the planning stage, and during the lesson, respectively.

For both the planning and during the lesson stages, many teachers focused on differentiating the tasks provided to students by the preparation of prompts, and by grouping arrangements. Interestingly, grouping suggestions of some teachers focused on mixed ability, while others suggested groups of similar ability. Differences between the comments in the two stages were the emphasis on careful choice of tasks and resources, taking into account the teachers' knowledge of individuals in relation to the content in the planning stage; and encouraging students to share their thinking, the development of a classroom culture, providing encouragement and enthusiasm, and monitoring students while they are working on the task, during the lesson.

Of course, there are some strategies which are more appropriately addressed during planning (e.g., choice of tasks), and during the lesson (e.g., the teacher monitoring students), respectively.

Primary teachers' insights after teaching up to ten tasks

In June, following the chance to try out up to 10 challenging tasks, teachers were given a verbal encouragement to provide only one thought, their most important change in practice that was different from the way they planned and taught previously. Tables 3 and 4 show the most frequently occurring categories. Of course, the request for just one response led to a smaller number of responses than earlier in the year. For this reason, percentages are not used here.

Possibly the most interesting difference in the data from before the teaching of the lessons (February) and after (June) was the emphasis on 'holding back'. In the June survey, 10 out of the 35 comments related to the teacher talking less, teaching less, or allowing students to struggle. Although this number is not large, it is important to remember that it represents a large proportion of responses out of 35.

Overall, 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 and work on the tasks.

It is worth commenting that an instruction to teachers to "hold back from telling" would be simplistic. While Hiebert and colleagues (1997) noted that "intervening too much and too deeply ... can easily cut off students' initiative and creativity, and can remove the problematic nature of the material" (p. 9), this does not imply a move away from any type of teacher telling and leaving students to flounder unnecessarily. For example, there are types of telling that stimulate students' mathematical thoughts "via the introduction of new ideas into a classroom conversation" (Lobato, Clarke & Ellis, 2005, p. 101). Finding the balance between allowing students to struggle with important mathematics and supporting their developing understanding is not easy. Nor is it easy to define in terms of what teachers should say or do.

Conclusion

From our experiences in using the tasks described in this article and in considering insights from teachers and the project team, we offer the following list of strategies for encouraging persistence on challenging tasks (Cheeseman, Clarke, Roche & Wilson, 2013; Sullivan, Clarke, Clarke & Roche, 2013):

* some attempt is made to connect the task with students' experience;

* 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.

Student persistence is important. As one student wrote in a written reflection:

"We do learn more when we're confused and we've got to work our way out of it."

Appendix

Missing Number Multiplication

I did a multiplication question correctly for homework, but my printer ran out of ink. I remember it looked like:

2 -- x 3 -- = -- -- 0

What might be the digits that did not print? (Give as many answers as you can)

Rationale for the lesson

Mathematics is fundamentally about patterns and the patterns can help us to understand key ideas. There are also important patterns in, for example, the numbers when multiplied that have an answer that ends in a 0.

Year level

Year 5-6

Particular pedagogical considerations

Explain that each line represents a digit (that is, the first number has 2 digits). As a first prompt for those who finish, ask whether they have all the possible answers. Encourage students to be systematic about how they record their answers.

For the students

Sometimes solving multiplication and division problems is about finding patterns. In this case the focus is on what numbers when multiplied have an answer that ends in 0.

Enabling prompt

Change the format of the calculation to the following:

-- x -- = -- 0

35 x -- = --0

Extending prompt

Change the format of the calculation to the following:

-- x -- 0 x 3 -- = -- -- 0

Consolidating task

I did a multiplication question correctly for homework, but my printer ran out of ink. I remember it looked like:

1 -- x 4 -- = -- 2

What might be the digits that that did not print? (Give as many answers as you can.)

Some possible student solution strategies

20 x 30 (or 31, or 32, or 33, and so on up to 39) will work So will 20 (or 21, or 22, or 23, and so on up to 29) x 30 Then 25 x 32 (or 34, or 36 or 38) will work Likewise 22 (or 24, or 26, or 28) x 30 will work

References

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.

Hiebert, J., Carpenter, T. P, Fennema, E., Fuson, K. C., Wearne, D., Murray, H., Olivier, A. & Human, P (1997). Making sense: Teaching and learning mathematics with understanding. Portsmouth, NH: Heinemann.

Lobato, J., Clarke, D. & Ellis, A. (2005). Initiating and eliciting in teaching: A reformulation of telling. Journal for Research in Mathematics Education, 36(2), 101-136.

Pogrow, S. (1988). Teaching thinking to at-risk elementary students. Educational Leadership, 45(7), 79-85.

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 & National Council of Teachers of Mathematics.

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

Sullivan, P (2011). Teaching mathematics: Using research-informed strategies. Australian Education Review 59. Camberwell, Victoria: Australian Council for Educational Research.

Sullivan, P, Cheeseman, J., Michels, D., Mornane, A., Clarke, D., Roche, A. & Middleton, J. (2011). Challenging mathematics tasks: What they are and how to use them. In L. Bragg (Ed.), Maths is multi-dimensional (Proceedings of the 48th Annual Conference of the Mathematical Association of Victoria, pp. 33-46). Melbourne: Mathematical Association of Victoria.

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.

Thompson, D. R., Battista, M. T., Mayberry, S., Yeatts, K. L. & Zawojewski, J. S. (2009). Navigating through problem solving and reasoning in Grade 6. Reston, VA: National Council of Teachers of Mathematics.

Anne Roche

Australian Catholic University

<anne.roche@acu.edu.au>

Doug Clarke

Australian Catholic University

<doug.clarke@acu.edu.au>

Table 1. Most common strategies in the planning stage for encouraging
persistence on challenging tasks.

Strategy for the    Number of   Percentage of all   Illustrative
planning stage      comments    comments (n=172)    comments

Differentiation        46             26.7%         * Make variations
                                                    to tasks to suit
                                                    the needs of the
                                                    children.

                                                    * Consider
                                                    extending/enabling
                                                    prompts.

Nature of tasks        25             14.5%         * Develop a task
                                                    that is open
                                                    ended.

                                                    * Careful task
                                                    selection.

Grouping               18             10.5%         * Ensure working
                                                    groups are mixed
                                                    ability.

                                                    * Group children
                                                    according to
                                                    ability.

Resources              18             10.5%         * Concrete
                                                    material.

                                                    * Plan and collect
                                                    all equipment
                                                    needed.

Teacher knowledge      18             10.5%         * Understand the
                                                    curriculum above
                                                    and below level.

--Content                                           * Be aware of
                                                    misconceptions.

Teacher knowledge      11             6.4%          * Understand
                                                    student learning
                                                    styles.

--Students                                          * Ensure s/he
                                                    knows where
                                                    students are at.

Table 2. Most common strategies during the lesson for encouraging
persistence on challenging tasks.

Strategy for during    Number of   Percentage of all   Illustrative
the lesson             comments    comments (n=164)    comments

Discussion/question-      38             23.2%         * Encouraging
                                                       students to
                                                       discuss
                                                       mathematics.

ing/ sharing                                           * Question
                                                       students to
                                                       investigate
                                                       their thinking.

Differentiation           21             12.8%         * Use enabling
                                                       prompts.

                                                       * Make changes
                                                       to the activity
                                                       to best suit
                                                       each child.

Grouping                  20             12.2%         * Allow
                                                       students to
                                                       work with a
                                                       partner to
                                                       share
                                                       strategies.

                                                       * Use flexible
                                                       groupings, kids
                                                       learn from each
                                                       other.

Culture                   16             9.8%          * Discuss
                                                       persistence
                                                       when it gets
                                                       tough.

                                                       * Reinforce
                                                       that taking
                                                       risks/making
                                                       mistakes is a
                                                       normal part of
                                                       learning.

Teacher enthusiasm/       13             7.9%          * Praise,
                                                       encourage
                                                       students by
                                                       focusing on
                                                       what they do
                                                       know.

encouragement                                          * Present
                                                       positively--
                                                       enthuse
                                                       students.

Teacher monitoring        13             7.9%          * Monitor
                                                       progress of
                                                       each
                                                       student/group
                                                       closely.

students                                               * Check in with
                                                       all students.

Table 3. Most common new strategies in the planning stage for
encouraging persistence.

Strategy in the   Number of         Illustrative comments
planning stage    teachers (n=35)

Differentiation         10          * Have the prompting questions
                                    already to use during the
                                    session, rather than waiting for
                                    a particular misunderstanding to
                                    occur.

                                    * I have planned what I will say
                                    to enable/challenge. This has
                                    been a change as previously I
                                    would do this as I am working
                                    with students on tasks.

Nature of tasks          7          * More problem solving
                                    activities. Plan more tasks that
                                    they need to think about instead
                                    of telling them what was wanted.

                                    * I would probably now give much
                                    harder tasks so that everyone had
                                    a level of confusion.

Holding back             3          * Not telling them what to do.

                                    * Not planning to 'teach' the
                                    concept first but waiting for the
                                    need to arise. Purposeful
                                    learning.

Table 4. Most common new strategies during the lesson for encouraging
persistence.

Strategy during   Number of         Illustrative comments
the lesson        teachers (n=35)

Discussion/             11          * Asking lots more questions;
questioning/                        e.g., So where could you go from
sharing                             there? Can you explain how you
                                    got here? What could you do next?
                                    Are you sure that's correct?

                                    * Students share more of their
                                    thinking more of the time.
                                    Students are learning more from
                                    sharing with each other, rather
                                    than listening to me.

Holding back            10          * I model less at the beginning
                                    of lessons.

                                    * I am more careful to hold back
                                    and not give the strategy which
                                    could help in the initial stage
                                    of the maths task.

Culture                  9          * I am a lot more willing to say
                                    to a student, "I know this is
                                    hard, I want it to be hard you
                                    need to go and think a bit more
                                    about (some specific context)"

                                    * Using phrases such as, "Yes
                                    this is hard", "Zone of
                                    confusion", "I want you to have a
                                    go first", "I'm not going to help
                                    you for 10 minutes", "Prove it to
                                    me", "How do you know it is
                                    correct?".