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Mathematical Knowledge for Teaching

514

Wood, M. B., Turner, E. E., Civil, M., & Eli, J. A. (Eds.). (2016). Proceedings of the 38th annual meeting of the

North American Chapter of the International Group for the Pyschology of Mathematics Education. Tucson, AZ:

The University of Arizona.

TEACHERS’ PERCEIVED DIFFICULTIES FOR CREATING MATHEMATICAL

EXTENSIONS AT THE BORDER OF STUDENTS’ DISCERNMENTS

A. Paulino Preciado-Babb Ayman Aljarrah Soroush Sabbaghan

University of Calgary University of Calgary University of Calgary

apprecia@ucalgary.ca aaljarra@ucalgary.ca ssabbagh@ucalgary.ca

Martina L. Metz Geoffrey G. Pinchbeck Brent Davis

University of Calgary University of Calgary University of Calgary

metzm@ucalgary.ca ggpinchb@ucalgary.ca abdavi@ucalgary.ca

In this paper we describe bonusing as a strategy to continuously extend students’ mathematical

understanding from tasks initially offered in the classroom. While this strategy resembles enrichment

activities often described in approaches based on Mastery Learning, there are fundamental

differences in our use of assessment and the nature of the tasks we propose. After two years of

implementing this strategy as part of the Math Minds initiative, we have found that teachers

perceived bonusing as relevant and engaging for students. However, they experienced difficulties in

the implementation of this strategy. In this paper, we draw from interviews with 14 elementary

teachers to identify difficulties they perceive while attempting to implement bonusing. We propose

the use of Variation Theory of Learning to inform the creation of bonus material and discuss

connections to knowledge for teaching.

Keywords: Elementary School Education, Mathematical Knowledge for Teaching, Instructional

Activities and Practices

Introduction

At the Math Minds initiative, a partnership that aims to improve mathematics instruction at the

elementary level (Metz, Sabbaghan, Preciado Babb, & Davis, 2015), we have come to perceive

bonusing as a strategy that fosters both a positive attitude towards mathematics and a deep

mathematical understanding. However, teachers in this initiative have consistently indicated that

creating and posing bonuses is challenging (Preciado Babb, Metz, Sabbaghan, & Davis, 2015). In

this paper we extend our findings on these challenges and propose that Marton’s (2015) Variation

Theory of Learning could inform the creation of bonus material. We also discuss implications for the

development of educative curricular material (Davis & Krajcik, 2005) that supports the learning of

both students and teachers.

Similar to the enrichment activities in Mastery Learning (Guskey, 2007), the teachers’ guides

used in this initiative advise teachers to “be ready to write bonus questions on the board from time to

time during the lesson for students who finish their quizzes or tasks earlier” (Mighton, Sabourin, &

Klebanov, 2010, p. A-8). Suggested strategies include: changing to larger numbers or introducing

new terms or elements; asking students to correct mistakes or complete missing terms in a sequence;

varying the task or the problem slightly; looking for applications of the concept; and looking for

patterns and asking students to describe them. In this paper we use the broader term bonus ‘tasks’—

instead of ‘questions’—to include activities that are not necessarily questions. We concur with

Bloom (1968) that these extension activities, or bonus tasks, should not just be a way to keep

students busy while supporting other students. We believe that bonusing has a positive effect on

students’ attitudes and dispositions towards mathematics. Teachers are encouraged to pose bonus

questions for all students, including those who perform lower in class, as suggested by Mighton

(2007), the developer of the resources used in the Math Minds initiative.

Mathematical Knowledge for Teaching

515

Wood, M. B., Turner, E. E., Civil, M., & Eli, J. A. (Eds.). (2016). Proceedings of the 38th annual meeting of the

North American Chapter of the International Group for the Pyschology of Mathematics Education. Tucson, AZ:

The University of Arizona.

We contend that the Variation Theory of Learning (Marton, 2015) has the potential to inform the

selection or the creation of bonus tasks. Example spaces, as defined by Watson and Mason (2005),

with sufficient variation can prompt learners to distinguish critical features of mathematical entities.

For Marton “the secret of learning is to be found in the pattern of variance and invariance

experienced by learners” (p. xi). Such patterns allow the learner to discern critical features required

to learn a particular concept or skill.

Methods and Data Sources

Our data for this report were drawn from interviews conducted in 2015 with 14 teachers, as well

as data from documented weekly classroom observations at two urban elementary research schools.

The interview transcripts were analyzed with a focus on teachers’ difficulties implementing

bonusing—motivated by early research results showing that teachers found bonusing to be quite

challenging (Preciado Babb et al., 2015). The research team, comprising two graduate students and

four researchers, met every week to discuss and analyze data.

Findings

By comparing teachers’ perceived difficulties from the interviews with our classroom

observations, the following four categories of difficulty were identified.

Lack of experience and knowledge

Many teachers agreed that once they became more familiar with the resource, bonusing would

become a less challenging task. The following excerpt is similar to many other teachers’ comments:

And just with working with you on bonusing and I think that the practice that I’ve had will

definitely help me next year. I don’t think next year will be as much of a struggle for me.

We documented many instances where students were asked to create their own bonuses. The

interviews revealed that, in some cases, students created questions were difficult for the teacher to

answer, as evidenced in the following comment:

Usually if it’s bonusing and students create their own bonusing or suggest something that is

beyond the curriculum there have been times where I’m like, “I don’t know the answer to that,”

and I don’t know…how should I answer that question? Do I set them up to answer it on their

own, or do I show them so that they don’t get confused? That’s when I hesitate. Usually with

the JUMP questions I should know how to answer it because the teacher’s book leads you.

The challenge of not being able to answer a student’s proposed bonus could be due to the

teacher’s lack of mathematical knowledge or to a disposition that makes him or her reluctant to

engage in unfamiliar situations. The last sentence of the excerpt suggests that the teacher relied on

the “teacher’s book” to deal with questions from the resource, reinforcing the idea that the difficulty

was due mainly to a lack of mathematical knowledge.

Difficult to bonus without going beyond the concept to be learned

Some teachers found it difficult to bonus without moving away from the goal of the lesson or the

concept to be learned. In the next excerpt, a teacher commented on the students who usually finish

early and/or who understand the content of the lesson faster than others:

They know they’re doing well and they know what’s coming next; they can anticipate that. And

so, it’s tricky for me to bonus them without moving away from the goal of the lesson. So, to give

them bonuses that challenge them and that they find exciting without changing the curriculum for

them.

Mathematical Knowledge for Teaching

516

Wood, M. B., Turner, E. E., Civil, M., & Eli, J. A. (Eds.). (2016). Proceedings of the 38th annual meeting of the

North American Chapter of the International Group for the Pyschology of Mathematics Education. Tucson, AZ:

The University of Arizona.

When teachers were asked to provide suggestions to the research team for improving the

initiative, several teachers insisted that they needed more pre-made bonus tasks that they could

implement in their classrooms:

Anything that you can sort of give us that’s like ready-made or ready to go on. … And bonusing

is a tricky thing that you have to, I guess, get your head around.

This particular difficulty based on the goal of the lesson relates to the range of students’ ability in the

classroom, as explained in the next paragraph.

Difficult to be responsive and improvise in bonusing

Most of the teachers asked for “ready at hand” bonuses because they didn’t know how to

integrate flexible bonusing in their plans. They claimed that this was due in part to the wide range of

ability between students. Teachers found it very difficult to create bonuses in the class or to modify

planned bonuses in response to classroom situations. This is evident in the following:

Just ’cause of the variety of students. And I’m not as confident in making those bonusing ‘cause

… when I think I’m bonusing with a small step, I find that I’m bonusing with a big step, and the

kids look at me with blank faces and I’m like, “Okay, what did I do wrong there?” I think it puts

the teacher in a tough position ’cause you have to bonus for each individual student and it’s

almost like you need to do it right there on the spot because if you plan it you have no idea.

In the previous excerpt, the teacher acknowledged that some bonus tasks turned out to be very

difficult for the class, even though the teacher was “bonusing with a small step.” Similar to the

requests for ‘ready-made’ bonus questions, some teachers complained about the lack of bonus

questions in the teachers’ guides for bonusing at a level appropriate for the students.

Limited time to create and implement bonusing

Sometimes teachers indicated time constraints that made it difficult to create and implement

bonusing. For example:

When it [the teacher’s guide] says, ‘Advanced’ or ‘Bonusing’ in the extra questions: They’re nice

for bonusing those students that need that type of bonus. For the class that I’m teaching right

now, I find that I’m not using that as much. It’s something that we don’t get to just in terms of

pacing but it is nice to have there just in case.

In this excerpt, the teacher acknowledged that students were not challenged with bonuses. This claim

contrasts with the explicit directive of the resource to create bonuses, as well as with the shared

perceptions from other teachers that bonusing is an important part of the program.

Time to prepare bonus material was a concern for some teachers. We observed that some of

them made efforts to create bonus questions attractive to students:

I find that just due to level of needs in my class sometimes I don’t—I find myself struggling with

the time to get around to them. And there’s such a range of students that I find that being able to

prepare ahead of time’s not always a reality if you’re planning five other subjects for the day.

It is not clear whether this teacher was creating the bonuses or just planning to use bonus tasks from

the resources. In either case, planning for bonusing was perceived as time consuming.

Discussion

In the difficulties described above we perceived a need for particular knowledge required to

create bonus material. Teachers consistently requested more ready-to-use bonus questions and even

teachers with two years or more in the initiative indicated a need to improve the way they create

bonus tasks.

Mathematical Knowledge for Teaching

517

The University of Arizona.

The difficulty in creating bonus tasks in response to students’ range of ability also supports the

idea that there is a particular knowledge teachers require for bonusing. We believe that a collection of

ready-to-use bonus material, either from the resource or from the research team, is neither practical

nor sufficient. In order to be responsive to students’ needs in the classroom, teachers should be able

to create bonus tasks on the spot with ease. Being able to improvise according to how the class

unfolds would also reduce the time constraints for preparation.

Our analysis also revealed a particular perception of bonusing as something that has to be done at

the end of the class, and only if time allows it. This perception of bonusing is restricting and might

entail additional time for preparation.

We propose that Marton’s (2015) Variation Theory of Learning holds great promise for

supporting teachers in the creation of bonus tasks. Bonus tasks may be created by varying the

original task in ways that allow new explorations and discoveries within the same concept or topic in

a lesson, or in a way that represents an opportunity to apply (generalize) the same principle to

different cases. Therefore, a future venue of research is how an understanding of this theory might

inform teachers’ implementation of bonusing in their classrooms.

We conclude with a potential implication for the development of and research on curricular

material. Just as educative curriculum materials (Davis & Krajcik, 2005) can support teachers’

learning, we believe that resources that teachers use in their classroom can facilitate the creation of

bonus tasks and thereby further impact their knowledge for teaching mathematics. For instance, we

have found that many tasks in the resources used in this initiative are presented in sequences of items

with a clear pattern of variation, and very often with explicit suggestions for bonuses. Teachers can

learn from the resource by paying attention to the patterns of variation related to key mathematical

concepts and ideas in the examples and tasks presented to students. However, we need to better

understand what teachers need to know to take advantage of the educative materials and what

features the educative materials need to embody to draw teachers’ attention to key mathematical

ideas.

References

Bloom, B. (1968). Learning for mastery. Evaluation Comment, 1(2), 1–12.

Davis, E. A., & Krajcik, J. S. (2005). Designing educative curriculum materials to promote teacher learning.

Educational Researcher, 34(3), 3–14.

Guskey, T. R. (2007). Closing achievement gaps: Revisiting Benjamin S. Bloom’s “Learning for Mastery.” Journal

of Advanced Academics, 19(1), 8–31. doi:10.4219/jaa-2007-704

Marton, F. (2015). Necessary conditions of learning. New York: Routledge.

Metz, M., Sabbaghan, S., Preciado Babb, A. P., & Davis, B. (2015). One step back, three forward: Success through

mediated challenge. In A. P. Preciado Babb, M. Takeuchi, & J. Lock (Eds.), Proceedings of the IDEAS 2015:

Design Responsive Conference (pp. 178-186). Calgary, Canada: Werklund School of Education. Retrieved from

http://dspace.ucalgary.ca/handle/1880/50872

Mighton, J. (2007). The end of ignorance: Multiplying our human potential. Toronto: Alfred A. Knopf.

Mighton, J., Sabourin, S., & Klebanov, A. (2010). JUMP Math 1 Teachers’ resources: Workbook 1. Toronto,

Ontario: JUMP Math.

Preciado Babb, A. P., Metz, M., Sabbaghan, S., & Davis, B. (2015). Insights on the relationships between

mathematics knowledge for teachers and curricular material. In T. G. Bartell, K. N. Bieda, R. T. Putnam, K.

Bradfield, & H. Dominguez (Eds.), Proceedings of the 37th annual meeting of the North American Chapter of

the International Group for the Psychology of Mathematics Education (pp. 796 – 803). Lansing, MI: Michigan

State University.

Watson, A. & Mason, J. (2005). Mathematics as a constructive activity: Learners generating examples. Mahwah,

NJ: Erlbaum.