Improving Mathematics Instruction Through the Use of Divergent Questions Alison Geisler Senior ...

18 Pages·2013·0.35 MB·English

by  Alison Geisler

Improving mathematics instruction 1
Improving Mathematics Instruction Through the Use of Divergent Questions
Alison Geisler
Senior Research: EN498-01
April 29, 2005
Improving mathematics instruction 2
Are elementary teachers asking ‘good questions’ in their mathematics lessons? Do
‘good questions’ elicit more student participation than other questions? The questions
teachers pose to students help to shape the quality of oral communication that takes place
in the classroom. Communication, a vital part of mathematics, can help students to
construct meaning. Research in mathematics instruction suggests that it is best learned
when constructed by the learner. Good questions in mathematics instruction, therefore,
are those that aid in students’ construction of meaning.
Communication in Mathematics
In their Principles and Standards for School Mathematics (2000), the National
Council of Teachers of Mathematics (NCTM) lists Communication, along with Problem
Solving, Reasoning and Proof, Connections, and Representation, as one of the five
essential Process Standards in mathematics. According to the Communication standard,
students should be able to communicate their mathematical ideas coherently using
mathematical language and be able to use communication as a way to organize their own
mathematical thinking. Students should also be able to evaluate the mathematical
thinking of others. These Standards recommend that teachers promote discussion in
which students give thoughtful responses, justify their answers, and rely on their own
mathematical reasoning to build their own understanding of mathematics (Mewborn &
Huberty, 1999; Rowan & Robles, 1998). Communication skills in mathematics are not
only essential for students to develop, but communication is an essential part of the
mathematics learning process.
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Constructivism in Mathematics Instruction
Students learn mathematics best when they construct their own mathematical
knowledge. What does it mean for students to construct meaning? The growth of
constructivism, the educational philosophy of constructing meaning, can be attributed to
the work of psychologists William Brownell, Jean Piaget, Jerome Bruner, and Zoltan
Dienes, all of whom have applied this philosophy to mathematics instruction.
Constructivism is the idea that meaningful learning is active rather than passive in that
students cannot merely receive knowledge, they must create its meaning. Construction of
meaning is a social process in which students reflect on their thinking and the thinking of
others in order to find patterns and relationships and to make generalizations. This
reflection happens through discussion (Reys, Lindquist, Lambdin, Smith, & Suydam,
2004). Constructing knowledge requires students to go beyond memorizing facts. It
requires students to take in new information, make sense of it in their minds, and then
integrate the new knowledge and their prior knowledge (Anderson et al., 2001).
In mathematics, students construct meaning when they actively participate in
lessons: manipulating objects, thinking aloud, sharing ideas, and making connections
between mathematical ideas. According to NCTM publications, mathematics instruction
should engage students’ intellect, provide opportunities for deepening understanding, and
organize classroom discussion to promote investigation and growth (Kennedy, Tipps, &
Johnson, 2004). One way to encourage this type of meaning construction, and to guide
student discussion that facilitates such construction, is through good questions.
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Good Questions in Mathematics Instruction
Just what are these ‘good questions’? To use the word ‘good’ is fairly vague. In
light of what is known about effective mathematics instruction, we can develop a more
precise description of a good question. Questions that help students to construct their own
understanding of mathematics, as discussed in the previous section, would certainly come
under this ‘good’ category. Weiss and Pasley (2004) define good questions in
mathematics as those that monitor students’ understanding and encourage them to think
more deeply about the content. Rowan and Robles (1998) describe good questions as
those that encourage students to use their prior knowledge to construct new
understanding. Schifter and Riddle (2004) advocate a classroom in which mathematical
ideas are explored rather than memorized and in which students communicate their
reasoning for deeper understanding. Good questions include those that call for students to
explain their thinking and reasoning (Rowan & Robles, 1998; Mewborn & Huberty,
1999). These ideas about good questions are supported by NCTM, which recommends
that questions appeal to students’ prior knowledge and encourage students to
communicate about their reasoning (Rowan & Robles, 1998). According to Reys et al.
(2004), good questions are those that promote critical thinking and meaningful
connections. The following are some examples of ‘good questions’:
How did you get your answer?
What do you think would happen if…?
How are … and… similar/different?
What pattern do you see?
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Why do you think that works?
Will it be the same if we use different numbers? Why or why not?
How do you think we should find out?
How do you know?
Have you ever solved a problem like this before?
(Reys et al., 2004; Rowan & Robles, 1998; Schielack, Chancellor, & Childs, 2000).
When we look at what these ‘good questions' have in common, we see that they are all
open-ended questions, or divergent questions, that invite more than one acceptable
answer.
Divergent and Convergent Questions in Mathematics Instruction
NCTM recommends that questions in mathematics instruction be open-ended
whenever possible (Rowan & Robles, 1998). Open-ended questions are also known as
divergent questions. Divergent questions invite multiple responses, while convergent
questions have one correct answer. An example of a divergent question in mathematics is,
“Why did you decide to use this equation?” It elicits varied responses from students, in
that different students will have different and equally valid answers to the same question.
Closed questions are also known as convergent questions. An example of a convergent
question in mathematics is, “What is the sum of two and four?” This question has one
correct answer. Both divergent and convergent questions are valuable in that they serve
different purposes. At certain times, teachers must ask for specific answers and so
convergent questions are used. However, divergent questions are useful in that they allow
teachers to more clearly see the mental processes of their students. Divergent questions
also encourage metacognitive thinking and are essential to constructing meaning in
mathematics learning.
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Divergent questions enable teachers to further the meaning-making process of
constructivism in the classroom. These questions spark students’ metacognitive processes
and require them to communicate about their reasoning. Divergent questions include
higher level questions, as defined by Bloom’s taxonomy of cognitive thinking, that
require students to understand, apply, analyze, evaluate, and create using their
knowledge. To go beyond merely regurgitating mathematical facts, these higher level
cognitive processes must take place (Anderson et al., 2001; Barton, 1997). Higher level
questions ensure that mathematics is more than just facts and algorithms to memorize and
repeat (Schifter & Riddle, 2004). Meaningful learning means students are able to transfer
their knowledge and process skills to new situations. This is problem-solving (Anderson
et. al., 2000).
Not only do divergent questions challenge students to think more deeply about the
content, but they encourage all students to contribute their ideas and become more
confident and independent in their learning (Rowan & Robles, 1998; Weiss & Pasley,
2004). In addition, students are more likely to volunteer answers to divergent questions
than to convergent questions (Martin, 2003), and this means that more students are
actively involved in the lesson. In other words, students are more willing to volunteer
answers to questions when there are many acceptable answers rather than one correct
answer. In looking at the types of questions teachers ask in math it can be deduced that
the ‘good questions’ that we hope teachers are asking tend to be divergent, or open-
ended, questions.
Improving mathematics instruction 7
Need for Improvement in Mathematics Instruction
There is evidence to suggest that mathematics instruction could be improved, and
that much of this improvement could come in the form of better questioning. A study of
364 Kindergarten through 12th grade math and science lessons found that only 15% of the
lessons were high in quality. Fewer than 20% contained elements such as effective
questioning, defined by the study as questions that monitor understanding and encourage
deeper thinking. Additionally, most questions used by the teachers were low-level,
convergent questions asked in rapid succession (Weiss & Pasley, 2004). In another study,
teachers reported that they ask twice as many convergent questions as divergent questions
(Harris, 2000).
An investigation of twelve teachers (Martin, 2003) found that 85% of questions
asked by the teachers were convergent and that such convergent questions received
shorter responses from students. These responses contained less mathematical
terminology than responses to divergent questions. Convergent questions also received
fewer ‘hands up,’ or students volunteering to answer the question, than when divergent
questions were asked. When placed into categories according to Bloom’s Taxonomy, 79%
of questions were Knowledge, Comprehension, or Application (lower level questions)
while only 21% were Analysis or Synthesis (higher level questions). There were no
observed questions on the Evaluation level. Most surprising, Martin interviewed the
teachers and found that none had prepared any of their questions in advance, or had even
thought about doing so.
Improving mathematics instruction 8
Methodology
In light of the available research on questioning in mathematics instruction,
including the effectiveness of good questions in helping students to construct meaning in
mathematics and the need for improved questioning in mathematics instruction, an
observation of local elementary mathematics lessons was conducted to record the
questions teachers ask and compare the number of divergent and convergent questions.
Additionally, the number of raised hands, or students volunteering to answer each
question, was recorded so that any relationship between the type of question asked and
the willingness from students to respond could be analyzed. This research was conducted
in hopes that more encouraging results would be found than those in the study conducted
by Martin (2003). Specifically, are teachers asking fewer divergent questions than
convergent ones in their math lessons? What proportion of questions teachers ask in
elementary math lessons require higher-level thinking?
Observations were conducted in three elementary classrooms, one each from the
first, third, and fourth grades, at Woodward Avenue Elementary School in Volusia
County, Florida. An observation instrument to aid in data collection (Appendix A) was
developed and implemented. Each oral question the teacher asked was recorded along
with the number of raised hands each question received. Later, each question was
categorized as either a divergent or a convergent question. As the data was being
gathered, the need for a third category, non-academic questions, became apparent and
was added.
Improving mathematics instruction 9
After collecting data from these lessons, the percent of divergent, convergent and
non-academic questions of the total number of questions asked in each lesson was found.
Pie charts for each lesson were created to represent this data. The data was then analyzed
to see what kind of relationship existed between the type of question asked and the
number of raised hands. For each type of question, the average number of hands raised in
response to that type of question was found and then was converted into a percent of the
total number of students present. Circle graphs were created to show the average percent
of students who raised their hand to answer each type of question for each lesson
observed.
In all cases, questions that were asked by the teacher that were answered chorally
by the whole class or that were asked specifically to individual students were taken out of
the data analysis, since students were not given the opportunity in either case to raise
their hands to answer these questions.
Results
A total of twenty-four oral questions were observed in the mathematics lesson
from the first grade. Of these questions, one was divergent, 22 were convergent, and one
was non-academic. In other words, 2% of the questions were divergent, 92% of the
questions were convergent, and another 2% were non-academic. Twelve and a half
percent of the students present for the lesson raised their hands to answer the divergent
question, an average of 25% raised their hands for the convergent questions, and an
average of 31.25% of the students raised their hands for the non-academic question.
Improving mathematics instruction 10
In the third grade lesson, 45 questions were observed. Of these questions, 18 were
divergent, 21 were convergent, and six were non-academic. In other words, 40% of the
questions were divergent, 47% of the questions were convergent, and 13% were non-
academic. An average of 25% of the students present raised their hands to answer the
divergent questions, an average of 28% raised their hands for the convergent questions,
and an average of 26% of the students raised their hands for the non-academic questions.
Thirty-three questions were observed in the fourth grade lesson. Of these
questions, 10 were divergent, 12 were convergent, and eleven were non-academic. In
other words, 30% of the questions were divergent, 37% of the questions were convergent,
and 33% were non-academic. An average of 11% of the students raised their hands to
answer the divergent questions, an average of 12.25% of the students raised their hands
for the convergent questions, and an average of 9.6% raised their hands for the non-
academic questions.
Data tables and graphs of the data from all three mathematics lessons can be
found in Appendices B through D.
Analysis of Results
In each case, the percent of convergent questions asked was greater than the
percent of divergent questions asked. This result is consistent with the literature that
reports a higher percentage of convergent questions being asked than divergent questions.
As the grade level increased, the percent of convergent questions asked decreased. A
significant percentage of the questions asked in the third and fourth grade lessons were
non-academic in nature. In each grade level, the student response to convergent questions