Task Design Following the Teachers’ Point of View.

Linda G. Opheim

Supervisors: John Monaghan and Simon Goodchild

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Background and Starting Point

Research project at UiA: Mathematics Tasks and Pupils Learning (MTPL):

• Vocational schools

• Task design

Personal starting point:

• New curriculums are not implemented as intended (Breiteig & Goodchild, 2010).

• Research is not easily transferred to the classrooms (Artigue, 2008; Breiteig &

Goodchild, 2010).

• Could this indicate a gap between researchers and teachers?

• Personal experience.

Previous Research – Beliefs

• There is no clear agreement about the definition of beliefs (Philipp, 2007).

• Different researchers have different perspectives given their ontological

foundation:

• Beliefs as a relative stable entity (Rogers, Cross, Gresalfi, Trauth-Nare, & Buck, 2011)

• Patterns of participation (Skott, Larsen, & Østergaard, 2011)

• «Agnostic» position (Schoenfeld, 2011)

• Research on teachers’ beliefs has mostly been preoccupied with identifying

beliefs and how these might hinder implementing reforms, and as a

consequence focused on how to change teachers’ beliefs (Philipp, 2007).

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Previous Research – Tasks

• A focus on what types of tasks the students will learn the most from.

• The Mathematical Task Framework a taxonomy of mathematical tasks with

respect to their cognitive demands (Stein, Smith, Henningsen & Silver 2000).

• Tasks with high levels of cognitive demands are difficult to implement for the

teachers (Henningsen & Stein, 1997; Stein & Lane, 1996).

• I have not found any research focusing on the teachers’ perspective when it

comes to task design.

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Previous Research – Tasks Cont.

• Math Wars (Becker & Jacob, 1998; Schoenfeld, 2004).

• Brown and McIntyre (1993) have conducted research where they investigate

teachers’ perspective on teaching. One of their conclusions about the

difficulties of implementing new innovations is that:

For the innovation to be ‘practical’, however, it would have to be so

clearly superior to the established practices, and so certainly achievable

and safe, as to justify the abandonment of the extensive repertoire of

teacher tactics, and the even more extensive craft knowledge about

when to use what tactics, that each teacher had built up over the years

(Brown & McIntyre, 1993, p. 116).

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Research Questions

• What is the relationship between teachers’ beliefs and the characteristics of

tasks these teachers prefer to use in the classroom?

• What characteristics of tasks are exposed in the textbooks?

• What are the differences in characteristics in textbook tasks and tasks that are

based on teachers’ beliefs?

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How can I Answer my Research Questions?

• RQ1: What is the relationship between teachers’ beliefs and the characteristics

of tasks these teachers prefer to use in the classroom?

• It is not given that a teacher’s beliefs are elaborated into fully articulated

philosophies (Ernest, 1989).

• Theory versus practice.

• Work with teachers with different beliefs.

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Design Cycle

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Discussion with the teacher about what kind of task he wants.

Design a first draft of the tasks.

Present and discuss the tasks with the teacher.

Refine the tasks.

Observe implementation in the classroom.

Evaluate the implementation together with the teacher.

Data Sources

• Semi-structured interviews with the teachers

• Documentation of design process together with the teachers

• The tasks

• Observing implementation

• Evaluation of the implementation

• Reflection notes

• Field notes

• Written communication with the teachers

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Design-based Research

• Design-based research is not so much an approach as it is a series of

approaches, with the intent of producing new theories, artifacts, and practices

that account for and potentially impact learning and teaching in naturalistic

settings (Barab & Squire, 2004, p. 2).

• Design research is a label that is used for a family of different research

approaches, for instance: design studies, design experiments, developmental

research, formative research and engineering research (van den Akker, Gravemeijer,

McKenny, & Nieveen, 2006).

• However, the design research community "have by no means reached

consensus on terminology or the warrants for their work" (Shavelson, Phillips, Towne,

& Feuer, 2003, p. 25).

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Design-based Research Cont.

• Some characteristics of design-based research:

• interventionist

• iterative

• process oriented

• utility oriented

• theory oriented

(van den Akker et al., 2006)

• My research is within the interpretive paradigm.

• The focus in my research is on teaching and not learning.

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Selecting the Teachers

• Passionate about teaching and mathematics.

• Two ‘traditional’ and two ‘reform oriented’.

• Contacted a teacher with a broad network.

• This teacher gave me names and made first contact on my behalf.

• Ethical challenge.

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Data Collected.

Semi-

structured

interview

Observation

in class

Other

communication

Design cycles Recordings in

the classroom

Recordings of

discussions

Total

Teacher 1 x 2 x 2 * 0,5 0 6h and 14min 6h and 14min

Teacher 2 x 1 x 1,5 59 min 2h and 51min 3h and 50min

Teacher 3 x 1 x 4 6h 9h and 30min 15h and 30min

Teacher 4 x 1 x 2 1h and 12min 3h and 50min 5h and 2min

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What Does the Teachers Request When it Comes to Tasks?

• Motivating

• Tasks which get the students working without having to explain.

• Tasks which makes the pupils understand through solving the task.

• Task which give the students Aha!-moments.

• Tasks which keep the students working.

• Tasks which are related to ‘real-life’ (utility).

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How do I Present Tasks?

• Give several alternatives.

• I try to present an idea - not a finished task.

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Some Limitations to the Tasks

• The teacher wants them to learn a rather limited and specific topic.

• Time to design tasks can be limited.

• No, classroom discussions are not possible.

• No, they can’t do this, because we haven’t had about Pythagoras yet.

• I have never put people together in pairs to work, so the students would be

surprised by my change of pedagogics.

• I am not going to bring with me a computer do demonstrate stuff, even if I see

it could be somewhat useful. Because I am not so used to it, and when

problems arise I am unsure how to handle it.

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Some Examples of Tasks

• Area task first version

• Area task revised version

• Logarithms:

• Historical task

• Mixing pH

• Big, bigger, biggest

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Analyzing my Data

• Concurrently data reducing all of my data.

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Analyzing Beliefs

• Something similar to Moreira and Noss (1995)

• Use main categories relevant to beliefs

• Thoroughly analyze the semi-structured interviews using these categories.

• Isolate parts in the data reductions relevant to the categories and analyze.

• Write a summary of each category on every teacher.

• Let the teacher read and comment on the text.

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Analyzing Characteristics of Tasks

• Initially intended to use previously developed characteristics of tasks

Problematic

• Plan to use techniques from grounded theory (Glaser & Strauss, 1967).

RQ1: What is the relationship between teachers’ beliefs and the characteristics of

tasks these teachers prefer to use in the classroom?

I need to develop a unit of analysis

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Research Outcome

I believe this research will be a valuable resource in bridging the gap between

researchers and teachers, and help us when developing tasks and textbooks that

are more likely to be implemented as intended.

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References

• Artigue, M. (2008, April 21-April 25). Didactical Design in Mathematics Education. Paper presented at the Nordic research in

mathematics education, Copenhagen.

• Barab, S., & Squire, K. (2004). Introduction: Design-Based Research: Putting a Stake in the Ground. The Journal of the

Learning Sciences, 13(1), 1-14.

• Becker, J. P., & Jacob, B. (1998). Math War. ICMI Bulletin, 44.

• Breiteig, T., & Goodchild, S. (2010). The Development of Mathematics Education as a Research Field in Norway: An Insider's

Personal Reflections The First sourcebook on Nordic research in mathematics education: Norway, Sweden, Iceland, Denmark

and contributions from Finland (pp. S. 3-9). Charlotte, N.C.: Information Age Publishing.

• Brown, S., & McIntyre, D. (1993). Making sense of teaching. Buckingham: Open University Press.

• Ernest, P. (1989). Mathematics teaching: the state of the art. New York: Falmer Press.

• Glaser, B. G., & Strauss, A. L. (1967). The discovery of grounded theory : strategies for qualitative research. New York: Aldine

de Gruyter.

• Philipp, R. A. (2007). Mathematics Teachers' Beliefs and Affect. In F. K. Lester (Ed.), Second handbook of research on

mathematics teaching and learning (Vol. 1, pp. 2 b. : ill.). Charlotte, N.C.: Information Age.

• Henningsen, M., & Stein, M. K. (1997). Mathematical Tasks and Student Cognition: Classroom-Based Factors That Support and

Inhibit High-Level Mathematical Thinking and Reasoning. Journal for Research in Mathematics Education, 28(5), 524-549.

• Moreira, C., & Noss, R. (1995). Understanding Teachers' Attitudes to Change in a LogoMathematics Environment. Educational

Studies in Mathematics, 28(2), 155-176. doi: 10.2307/3482894

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References cont.

• Rogers, M., Cross, D., Gresalfi, M., Trauth-Nare, A., & Buck, G. (2011). First Year Implementation of a Project-based Learning

Approach: The Need for Addressing Teacher’s Orientations in the Era of Reform. International Journal of Science and

Mathematics Education, 9(4), 893-917. doi: 10.1007/s10763-010-9248-x

• Schoenfeld, A. H. (2004). The Math Wars. Educational Policy, 18(1), 253-286. doi: 10.1177/0895904803260042

• Schoenfeld, A. H. (2011). How we think: a theory of goal-oriented decision making and its educational applications. New York:

Routledge.

• Shavelson, R. J., Phillips, D. C., Towne, L., & Feuer, M. J. (2003). On the Science of Education Design Studies. Educational

Researcher, 32(1), 25-28.

• Skott, J., Larsen, D. M., & Østergaard, C. H. (2011). From beliefs to patterns of participation – shifting the research perspective

on teachers. NOMAD Nordisk Matematikkdidaktikk/Nordic Studies in Mathematics Education, 16(1-2), 29-55.

• Stein, M. K., & Lane, S. (1996). Instructional Tasks and the Development of Student Capacity to Think and Reason: An

Analysis of the Relationship between Teaching and Learning in a Reform Mathematics Project. Educational Research and

Evaluation: An International Journal on Theory and Practice, 2(1), 50 - 80.

• Stein, M. K., Smith, M. S., Henningsen, M. A., & Silver, E. A. (2000). Implementing standards-based mathematics instruction:

a casebook for professional development: Teachers College Press National Council of Teachers of Mathematics.

• van den Akker, J., Gravemeijer, K., McKenny, S., & Nieveen, N. (2006). Introducing educational design research. In J. van den

Akker, K. Gravemeijer, S. McKenny & N. Nieveen (Eds.), Educational design research (pp. XI, 163 s.). London: Routledge.

• Wagner, J. (1997). The Unavoidable Intervention of Educational Research: A Framework for Reconsidering Researcher-

Practitioner Cooperation. Educational Researcher, 26(7), 13-22.

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