T

Task-Based Interviews inMathematics Education

Carolyn A. Maher1 and Robert Sigley2

1Robert B. Davis Institute for Learning,

Graduate School of Education, Rutgers –

The State University of New Jersey,

New Brunswick, NJ, USA2Rutgers – The State University of New Jersey,

New Brunswick, NJ, USA

Keywords

Clinical interview; Teaching experiment;

Problem solving; Task design

Definition

Interviews in which a subject or group of

subjects talk while working on a mathematical

task or set of tasks.

The Clinical Interview

Task-based interviews have their origin in clinical

interviews that date back to the time of Piaget,

who is credited with pioneering the clinical inter-

view. In the early 1960s, the clinical interview

was used in order to gain a deeper understanding

of children’s cognitive development (e.g., Piaget

1965, 1975). Task-based interviews have been

used by researchers in qualitative research in

S. Lerman (ed.), Encyclopedia of Mathematics Education, D# Springer Science+Business Media Dordrecht 2014

mathematics education to gain knowledge about

an individual or group of students’ existing

and developing mathematical knowledge and

problem-solving behaviors.

Task-Based Interview

The task-based interview, a particular form of

clinical interview, is designed so that inter-

viewees interact not only with the interviewer

and sometimes a small group but also with a

task environment that is carefully designed for

purposes of the interview (Goldin 2000). Hence,

a carefully constructed task is a key component

of the task-based interview in mathematics

education (Maher et al. 2011). It is intended to

elicit in subjects estimates of their existing

knowledge, growth in knowledge, and also their

representations of particular mathematical ideas,

structures, and ways of reasoning.

In preparing a clinical task-based interview,

certain methodological considerations warrant

attention and need to be considered in protocol

design. These require attention to issues of

reliability, replicability, task design, and general-

izability (Goldin 2000). Some interviews are

structured, with detailed protocols determining,

in advance, the interviewer’s interaction and

questions. Other protocols are semi-structured,

allowing for modifications depending on the

judgment of the researcher. In situations where

the research is exploratory, data from the inter-

views provide a foundation for a more detailed

protocol design. In other, more open-ended

OI 10.1007/978-94-007-4978-8,

T 580 Task-Based Interviews in Mathematics Education

situations, a task is presented and there is minimal

interaction of the researcher, except, perhaps, for

clarification of responses or ensuring that the

subjects understand the nature of the task.

Methodology

As subjects are engaged in a mathematical

activity, researchers can observe their actions

and record them with audio and/or videotapes

for later, more detailed, analyses. The recordings,

accompanied by transcripts, observers’ notes,

subjects’ work, or other related metadata,

provide the data for analyses and further protocol

design. Data from the interviews are then coded,

analyzed, and reported according to the research

questions initially posed.

Techniques and Resources

A variety of techniques are used in task-based

interviews, such as thinking aloud and open-

ended prompting (Clement 2000). These can be

modified and adjusted, according to the judgment

of the researcher.

Task-based interviews are used to investigate

subjects’ existing and developing mathematical

knowledge and ways of reasoning, how ideas are

represented and elaborated, and how connections

are built to other ideas as they extend their

knowledge (Maher 1998; Maher et al. 2011).

Episodes of clinical, task-based interviews can

be viewed by accessing the VideoMosaic Collab-

orative, VMC, website (http://www.videomosaic.

org) or Private Universe Project in Mathematics

(http://www.learner.org/workshops/pupmath).

An example of a task-based interview in which

the interviewee is engaged with the interviewer as

well as the task environment that was designed by

the researchers, see http://hdl.rutgers.edu/1782.1/

rucore00000001201.Video.000062046. The epi-

sode shows nine-year-old Brandon, explaining

the notation he used to explain his reasoning. It

also shows how the interviewer’s intervention,

asking Brandon if the solution reminded him of

any other problem, prompted him, spontaneously,

to provide a convincing solution for an isomor-

phic problem (Maher and Martino 1998).

A second example from the content strand of

algebra is a task-based interview of Stephanie,

an 8th grade girl who has been asked to build

a model for (a + b)3 with a set of algebra blocks.

Stephanie, earlier in the interview, has success-

fully expanded (a + b)3 algebraically to the

expression a3 + 3a2b + 3ab2 + b3 and is challenged

by the researcher in this clip to find each of the

terms as it is modeled in the cube that she builds.

In this example, the researcher is assessing

Stephanie’s ability to connect her symbolic and

physical representations as well as observing

how she navigates the transition from a two-

dimensional model of (a + b)2 to a model that

involves three dimensions. All nine of the clips

from this interview are available on the Video

Mosaic Collaborative website and can be found

by searching for the general title: Early algebra

ideas about binomial expansion, Stephanie’s

interview four of seven. The full title of clip 5

is Early algebra ideas about binomial expansion,

Stephanie’s interview four of seven, Clip 5 of 9:

Building (a + b)3 and identifying the pieces. Thelink to this clip is http://hdl.rutgers.edu/1782.1/

rucore00000001201.Video.000065479.

Task-Based Interviews for Assessment

Paper and pencil tests are limited in that they do

not address conceptual knowledge and the pro-

cess by which a student does mathematics and

reasons about mathematical ideas and situations.

Adaptations of the clinical task-based interview

have been useful in describing student knowledge

and providing insight into how mathematical

solutions to tasks are built by students. By provid-

ing a structured mathematical task, researchers

can gain insight into students’ mathematical

thinking (Davis 1984). Also, teachers can use

task-based interviews in their classrooms to

study how young children think about and learn

mathematics as well as to assess the mathematical

knowledge of their students (Ginsburg 1977).

Assessments of the mathematical understanding

and ways of reasoning in problem-solving situa-

tions of small groups of students can also be made

with open-ended task-based assessments (Maher

and Martino 1996). See http://www.learner.org/

workshops/pupmath/workshops/wk2trans.html.

An example of a group interview facilitated by

researchers Carolyn Maher and Regine Kiczek

Task-Based Interviews in Mathematics Education 581 T

T

with four 11th grade students who have been

working on combinatorics problems as a part of

a longitudinal study of children’s mathematical

reasoning since they were in elementary school

(Alqahtani, 2011). In this session they were

discussing the meaning of combinatorial notation

and the addition of Pascal’s identity in terms

of that notation. They were asked to write the

general form of Pascal’s identity with reference

to the coefficients of the binomial expansion.

Their work during the session indicates their

recognition of the isomorphism between the

binomial expansion and the triangle and can be

viewed at http://videomosaic.org/viewAnalytic?

pid¼rutgers-lib:35783.

The Teaching Experiment

According to Steffe and Thompson (2000),

a teaching experiment is an experimental tool

that derives from Piaget’s clinical interview. In

this context, the interviewer and interviewee’s

actions are interdependent. However, it differs

from the clinical interview in that the interviewer

intervenes by experimenting with inputs that

might influence the organizing or reorganizing

of an individual’s knowledge in that it traces

growth over time. In a teaching experiment,

researchers create situations and ways of

interacting with students that promote modifica-

tion of existing thinking, thereby creating a focus

for observing the students’ constructive process.

There typically is continued interaction with the

student (or students) by the researcher who is

attentive to major restructuring of and

scaffolding growth in the student’s building of

knowledge. In these ways, the teaching experi-

ment makes use of and extends the idea of

a clinical interview.

Yet a teaching experiment is similar to a task-

based interview in several ways. First, a problem-

atic situation is posed. Second, as the interviewer

assesses the status of the student’s reasoning in

the process of interacting with the student, new

situations are created in the attempt to better

understand the student’s thinking. Also, as in

some task-based interviews, protocols may be

modified as observation of critical moments

suggests (Steffe and Thompson 2000).

Significance

There is substantial and growing evidence that

clinical task-based interviews and their variations

provide important insight into subjects’ existing

and developing knowledge, problem-solving

behaviors, and ways of reasoning (Newell and

Simon 1972; Schoenfeld 1985, 2002; Ginsburg

1997; Goldin 2000; Koichu and Harel 2007;

Steffe and Olive 2009; Maher et al. 2011). The

interviews provide data formaking students math-

ematical knowledge explicit. They offer insights

into the creative activity of students in

constructing new knowledge as they are engaged

in independent and collaborative problem solving.

Cross-References

▶ Inquiry-Based Mathematics Education

▶ Problem Solving in Mathematics Education

▶Questioning in Mathematics Education

References

Alqahtani M (2011) Pascal’s identity. Video annotation.

Video Mosaic Collaborative. http://videomosaic.org/

viewAnalytic?pid¼rutgers-lib:35783

Clement J (2000) Analysis of clinical interviews:

foundation and model viability. In: Lesh R, Kelly AE

(eds) Research design in mathematics and science

education. Erlbaum, Hillsdale, pp 547–589

Davis RB (1984) Learning mathematics: the cognitive

science approach to mathematics education. Ablex,

Norwood

Ginsburg, H. (1977). Children’s arithmetic: The learning

process. New York: Van Nostrand.

Ginsburg HP (1997) Entering the child’s mind: the clinical

interview in psychological research and practice.

Cambridge University Press, New York

Goldin G (2000) A scientific perspective on structures,

task-based interviews in mathematics education

research. In: Lesh R, Kelly AE (eds) Research design

in mathematics and science education. Erlbaum,

Hillsdale, pp 517–545

Koichu, B., & Harel, G. (2007). Triadic interaction in

clinical task-based interviews with mathematics

teachers. Educational Studies in Mathematics, 65(3),

349–365.

Maher CA (1998) Constructivism and constructivist

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Mathematics teaching from a constructivist point of

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T 582 Teacher as Researcher in Mathematics Education

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methods of proof: the gang of four. In: Nesher P, Steffe

LP, Cobb P, Greer B, Goldin J (eds) Theories of

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morphism. In: Maher CA (ed) Can teachers help

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the process, vol 5. Universidade Santa Ursula, Rio de

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ics and reasoning: representing, justifying and building

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Prentice-Hall, Englewood Cliffs

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Littlefield Adams, Totowa

Schoenfeld A (1985) Mathematical problem solving.

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Schoenfeld A (2002) Research methods in (mathematics)

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Springer, New York

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methodology: underlying principles and essential

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Hillsdale, pp 267–307

Teacher as Researcher inMathematics Education

Dany Huillet

Faculty of Sciences, University of Eduardo

Mondlane, Maputo, Mozambique

Keywords

Teacher as researcher; Teacher training; Teacher

knowledge

The term “teacher as researcher” is usually used

to indicate the involvement of teachers in educa-

tional research aiming at improving their own

practice. The teachers-as-researchers movement

emerged in England during the 1960s, in the

context of curriculum reform and extended into

the 1980s. Cochran-Smith and Lytle (1999)

reviewed papers and books published in the

United States and in England in the 1980s dis-

seminating some experiences of teacher research.

The main feature of the teacher research move-

ment during this period seems to be an “explicit

rejection of the authority of professional experts

who produce and accumulate knowledge in

“scientific” research settings for use by others in

practical settings” (1999, p. 16). Within this

movement, teachers are no longer considered as

mere consumers of knowledge produced by

experts, but as producers and mediators of knowl-

edge, even if it is local knowledge, to be used in

a specific school or classroom. This knowledge

aims at improving teaching practice.

In mathematics education worldwide, the

teachers-as-researchers movement has been the

subject of debate within the mathematics educa-

tors’ community and of several papers presenting

results of these programs or discussing certain

aspects of teacher research (see Huillet et al.

2011). In these debates, the contention pivoted

around whether its outputs could be regarded as

research. Many research endeavors conducted by

teachers do not fill the requisites of formal

research, such as systematic data collection and

analysis, as well as dissemination of the research

results. Some researchers distinguish two forms

of teacher research in practice: formal research,

aimed at contributing knowledge to the larger

mathematics education community, and less

formal research, also called practical inquiry or

action research, which aims to suggest new ways

of looking at the context and possibilities for

changes in practice (Richardson 1994). A major

aim of most action research projects is the genera-

tion of knowledge among people in organizational

or institutional settings that is actionable – that is,

research that can be used as a basis for conscious

action (Crawford and Adler 1996).

The International Group for the Psychology of

Mathematics Education (PME) started a working

group called “teachers as researchers” in 1988.

This group met annually for 9 years and

published a book based on contributions from

its members (Zack et al. 1997). The book