Formative Questioning in Mathematics: An Open or Closed Case Study? Introduction Despite the overwhelming success of Bloom’s Taxonomy of Learning Objectives (Anderson & Sosniak, 1994) and Black’s and Wiliams’ (1998) extensive research on formative assessment, questioning still requires development in the mathematics classroom “to check and probe understanding” (Ofsted, 2012, p.34-35). In the author’s school, questioning in mathematics has been identified as requiring improvement; this essay is a proposal for a micro-research study to empirically investigate both the question types and the questioning techniques which encourage mathematical thinking and participation with the aim of identifying effective questioning in this school and provide recommendations for improvement. In this study, question type refers to the mathematical thinking intended and questioning technique refers to the strategies that teachers put in place for learners to think about and respond to questions. A literature review of learning objective taxonomies and their limitations considers those which are suitable to identify question types and levels of complexity to probe mathematical understanding; a combination of the most relevant taxonomies discussed will be used to classify questions for the purpose of this study. A review of Black’s and Wiliam’s theoretical f ramework on formative assessment analyses techniques to support formative questioning in mathematics. Based on the conclusions of the literature review, research questions are presented on the question types and questioning techniques employed in this school. The research design and methodology is described in terms of a mixed-method approach of quantitative research methods, by means of lesson observations and learner questionnaires, and qualitative methods, using semi-structured interviews with teachers to give them a voice on their intentions. Ethical issues are considered and sampling implications and analysis rationale discussed, including strategies to increase the reliability and validity of the study. Literature Review Black et al. (2006) believe effective questioning is essential to develop metacognition and self-awareness, so learners “can ask questions of each other and the focus can move from the teacher to the pupils” (p.128). However, research shows that teachers’ questioning is “not always well judged or productive for learning” (DfES, 2004, p.4) and highlights the need to use “open, higher-level questions to develop pupils’ higher-order thinking skills” (ibid, p.18). In the 1950s and 60s there were many attempts to produce a hierarchy for the complexity of thinking skills (Gall, 1970), however it was Bloom’s (1956) Taxonomy that
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Formative Questioning in Mathematics: An Open or Closed Case Study?
Introduction
Despite the overwhelming success of Bloom’s Taxonomy of Learning Objectives (Anderson
& Sosniak, 1994) and Black’s and Wiliams’ (1998) extensive research on formative
assessment, questioning still requires development in the mathematics classroom “to check
and probe understanding” (Ofsted, 2012, p.34-35). In the author’s school, questioning in
mathematics has been identified as requiring improvement; this essay is a proposal for a
micro-research study to empirically investigate both the question types and the questioning
techniques which encourage mathematical thinking and participation with the aim of
identifying effective questioning in this school and provide recommendations for
improvement. In this study, question type refers to the mathematical thinking intended and
questioning technique refers to the strategies that teachers put in place for learners to think
about and respond to questions. A literature review of learning objective taxonomies and
their limitations considers those which are suitable to identify question types and levels of
complexity to probe mathematical understanding; a combination of the most relevant
taxonomies discussed will be used to classify questions for the purpose of this study. A
review of Black’s and Wiliam’s theoretical framework on formative assessment analyses
techniques to support formative questioning in mathematics. Based on the conclusions of
the literature review, research questions are presented on the question types and questioning
techniques employed in this school. The research design and methodology is described in
terms of a mixed-method approach of quantitative research methods, by means of lesson
observations and learner questionnaires, and qualitative methods, using semi-structured
interviews with teachers to give them a voice on their intentions. Ethical issues are
considered and sampling implications and analysis rationale discussed, including strategies
to increase the reliability and validity of the study.
Literature Review
Black et al. (2006) believe effective questioning is essential to develop metacognition and
self-awareness, so learners “can ask questions of each other and the focus can move from
the teacher to the pupils” (p.128). However, research shows that teachers’ questioning is
“not always well judged or productive for learning” (DfES, 2004, p.4) and highlights the need
to use “open, higher-level questions to develop pupils’ higher-order thinking skills” (ibid,
p.18). In the 1950s and 60s there were many attempts to produce a hierarchy for the
complexity of thinking skills (Gall, 1970), however it was Bloom’s (1956) Taxonomy that
experienced “phenomenal growth” (Bloom in Anderson & Sosniak, 1994, p.1) and became
widely accepted as the optimal classification (Gall, 1970).
Bloom et al. (1956) were aware of the limitation that the Taxonomy classifies observable
behaviours, so it is not explicit how learning is constructed; instead they hoped the
classification would contribute to the development of a more complete theory of learning.
However, educational changes occurred mostly at policymaker level rather than having direct
influence on teachers (Anderson & Sosniak, 1994). Other criticisms include the omission of
the term ‘understanding’ (Furst, 1994) and the hierarchy implying that ‘knowledge’ leads to
intellectual abilities (Bereiter & Scardamalia, 1999), which are addressed (Figure 1) in a
revised taxonomy (Anderson et al., 2001).
Figure 1 – Bloom’s Original and Revised Taxonomies (Image from ODU, 2013)
Interchanging the top two tiers of the hierarchy perhaps reflects the importance of student-
centred learning in 21st Century education and replacing the nouns with verbs, could indicate
active learner participation, however neither version are intended as a “constructive way of
planning and answering questions” (Morgan & Saxton, 2006, p.19), rather it is a framework
about knowledge so “helps us to see the kind of thinking we can set into action through
questions” (ibid).
Anderson et al., (2001), consider remembering and understanding to be lower-order thinking
skills, while applying, analysing, evaluating and creating are considered higher-order,
however mathematical understanding is not necessarily a linear progression (Sfard, 1991;
Gray & Tall, 1991). Watson (2007) believes that Bloom’s Taxonomy “does not provide for
post-synthetic mathematical actions, such as abstraction and objectification” (p.114) and that
it “underplays knowledge and comprehension in mathematics” (ibid) as these can be
interpreted at different levels of mathematical thought. Watson (2003) also criticises the
simplicity of open and closed questioning as opportunities to extend conceptual
understanding in mathematics are of greater importance.
An alternative taxonomy is Biggs’ and Collis’ SOLO (Structure of Observed Learning
Outcomes) which proposes a sequence of unistructural, multistructural and relational
understanding (Pegg & Tall, 2010, p.174), which Watson (2007) believes “can be used to
devise questions which make finer distinctions than the vague notions of ‘lower-order’ and
‘higher-order’” (p.115). However, while the SOLO model allows for mathematical
abstraction, Watson (2007) argues that what a teacher intends and what a learner perceives
do not necessarily agree.
Smith et al. (1996) agree that Bloom’s Taxonomy has limitations in mathematics and propose
the MATH Taxonomy (Mathematical Assessment Task Hierarchy) for constructing
examination questions (Figure 2).
Figure 2 – MATH Taxonomy (Smith et al., 1996, p.67)
This could be a possible way of analysing verbal questioning in mathematics as the groups
distinguish the hierarchy of different types of activity which require either a “surface
approach” (Smith et al, 1996, p.67) or “deeper approach” (ibid.), rather than a hierarchy of
difficulty.
Andrews et al. (2005) use seven mathematical foci to analyse teachers’ behaviour (Figure 3).
Figure 3 – Mathematical Foci (Andrews et al., 2005, p.11)
According to Watson (2007), these foci describe “the intentions of teaching through
classifying features of mathematical meaning and structure without assuming that learners
necessarily do what is intended” (p.116). If combined with the MATH hierarchy, these foci
could provide a useful framework for analysing mathematical questioning in classroom
discourse (Appendix 1).
There exist other structures which are designed specifically for classifying questioning. For
example Morgan and Saxton (2006) classify questioning in three ways: Probing what is
already known; building a context for shared understanding; and challenging students to think
critically and creatively, however the second category could contain a large array of question
types and levels of complexity. Another distinction is in product-process questioning (Mujis &
Reynolds, 2011), where the former is designed to find the result while the latter is focused
more on the procedure, however in mathematics process is not necessarily considered
Question Type (Intended Mathematical Thinking) Coding Table
QUESTION TYPE Adapted from
Smith et al. (1996) and Andrews et al.
(2005)
PROMPTS Adapted from Watson’s
analytical instrument(2007, p.119)
FORMATIVE QUESTION STEMS From Wiliam (2006)
SURFACE APPROACH QUESTION
CODING
DEEPER APPROACH QUESTION
CODING
Factual
Name
Recall facts
Give definitions
Define terms
FS FD
Procedural
Imitate method
Copy object
Follow routine procedure
Find answer using procedure
Give answer
PS PD
Structural
Show me…
Analyse
Compare
Classify
Conjecture
Generalise
Identify variables
Explore variation
Look for patterns
Identify relationships
Tell me about the problem. What do you know about the problem? Can you describe the problem to someone else? What is similar . . . ? What is different . . . ? Do you have a hunch? . . . a conjecture? What would happen if . . . ? Is it always true that . . . ? Have you found all the solutions?
SS SD
Reasoning
Justify
Interpret
Visualise
Explain
Exemplify
Informal induction
Informal deduction
Can you explain/ improve/add to that explanation? How do you know that . . . ? Can you justify . . .?
RS RD
Reflective
Summarise
Express in own words
Evaluate
Consider advantages/ disadvantages
What was easy/difficult about this problem . . . this mathematics? What have you found out? What advice would you give to someone else about . . . ?
VS VD
Derivational
Prove
Create
Design
Associate ideas
Apply prior knowledge (in new situations)
Adapt procedures
Find answer without known procedure
Have you seen a problem like this before? What mathematics do you think you will use? Can you find a different method? Can you prove that . . . ?
DS DD
Appendix 2
Questioning Technique Coding Table
QUESTIONING TECHNIQUE CODE
Use random methods to choose a student to answer (e.g. names from hat) R
Hands up H
No hands up with ‘wait time’ N
Discuss answer for a set time in pairs/groups first G
Use mini-whiteboards to write answers W
Generate discussion from mini-whiteboards W+D
Choose from a few answers (e.g. Using voting fans) V
Ask if a student agrees with another A
Identify the error M
Writing up selection of responses on board then discuss S
Odd one out O
Always/Sometimes/Never True (or equivalent) T
Problems with more (or less) than one correct solution P