A Critical Review of Writing in the Mathematics Classroom with Students with Learning Difficulties
Nara Riplinger
Coursework Project EDSP9012 Masters of Special Education
Flinders University March 2008
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TABLE OF CONTENTS
Page ABSTRACT iii DECLARATION iv Chapter 1. INTRODUCTION 1 Purpose of the Study 1
Rationale 2 Research Methods 4 Definitions 4 Limitations and Delimitations 5
2. LITERATURE REVIEW 6
Transmission Theory 6 Transmission Theory and Students with Learning Difficulties 7 Current Theoretical Impetus: Constructivist Theory 8 Constructivist Theory and Students with Learning Difficulties 10 Constructivism Through Writing 11 The Theory of Writing 12 Advantages and Drawbacks for Students 14 Advantages and Drawbacks for Students
with Learning Difficulties 20 Advantages and Drawbacks for Teachers 23 A Shift in Pedagogy from Assessment of Learning
to Assessment For Learning 26 Writing as a Tool for Assessment for Learning
in a Constructivist Classroom 27 3. IMPLEMENTING WRITING IN THE
MATHEMATICS CLASSROOM 30 When to Write 31 What to Write 32 Assessment Methods 34
4. CONCLUSION 35 5. REFERENCES 38 6. APPENDIX A: Possible Writing Prompts for Mathematics 8 44
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ABSTRACT
In this time of mathematics teaching reform, many external governing bodies are encouraging
teachers to rely less on explicit teaching methods and rote fact memorization and to focus more
on the process of understanding and appropriately applying mathematical concepts. In this
manner, it is hoped that more numerate students will emerge from the education system. Many
teachers, however, may be at a loss as to how to emphasize and assess conceptual understanding
in the mathematics classroom. This paper will introduce the pedagogy of writing in the
mathematics classroom. It will present the drawbacks as well as the benefits of writing for both
students with learning difficulties and their teachers. It will also investigate how teachers may
use journal writing to formatively assess student learning. Finally, strategies that math teachers
can employ will be presented.
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DECLARATION
I certify that this project does not incorporate without acknowledgement any material previously
submitted for a degree or diploma in any university; and that to the best of knowledge and belief
it does not contain any material previously published or written by another person where due
reference is not made in the text.
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ACKNOWLEDGEMENTS
Thanks are due to my family for their support in my completing this project and especially to my
father for his assistance and guidance throughout my Masters program.
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CHAPTER 1
Introduction Purpose of the Study
A numerate individual has the ability to effectively communicate the processes used in
mathematics and to appropriately apply these skills (Higginson, n.d.; Steen, 2000). As numeracy
moves to the forefront of many education systems, including British Columbia’s, how to best
teach numeracy is debated. While mathematics may be defined as a fairly abstract subject
involving proofs, formula, and rules, numeracy is the ability to appropriately apply mathematical
concepts (Manaster, 2000; Math for families, 2007; Steen, 2000). To help students become
numerate individuals, many teachers support the constructivist model whereby students
internalize knowledge by constructing their own understanding of concepts; however, this
acceptance is somewhat disjointed with actual classroom practice (Cobb, 1988). This paper will
explore how writing to learn in the mathematics classroom may support students to better
understand and communicate mathematical concepts. The literature review undertaken in this
study will attempt to answer the following questions:
1. What does current research suggest about writing to learn for all students?
2. How might the use of writing become a tool for assessment for learning in the
mathematics classroom?
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3. What processes can math teachers employ to have students with learning difficulties write
in the mathematics classroom?
Rationale
The BC Ministry of Education has included improving numeracy skills, along with literacy skills,
as educational goals for the province. Prescribed Learning Outcomes for every grade, as well as
Performance Standards for grades one to eight, have been developed. Prescribed Learning
Outcomes are grade-specific concepts and skills that teachers are responsible for including in the
classroom curriculum. For mathematics, the learning outcomes are very specific and fairly
lengthy. Grade 8 Mathematics, for example, includes a unit on fractions; one of the eight
learning outcomes from this unit is “It is expected that students will represent and apply
fractional percents and percents greater than 100 in fraction or decimal form and vice versa”
(Mathematics 8 & 9: Integrated Resource Package, 2001). While teachers try to fulfill their legal
obligation to teach and assess every learning outcome, the Ministry of Education has now
developed Performance Standards for voluntary use in BC schools (BC Performance Standards,
2002). The Performance Standards focus on performance assessment in which students apply
mathematical skills and concepts to complex, problem-solving tasks. The Performance Standards
are criterion-referenced and the evaluation procedures are shared with the students. Teachers can
use the standards to monitor, evaluate and report on student performance, provide students and
parents with feedback, and plan instruction.
While the performance standards are presented in a neat, easy to read, chart format, there is vast
white space in which “teachers can … plan instruction or assignments, tailor or elaborate the
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criteria for specific activities or students…” (BC Performance Standards, 2002). However,
between the Prescribed Learning Outcomes and the Performance Standards, very little is
suggested to teachers about how to specifically reach these goals. Goldman and Hasselbring
(1997) argue that specifying new standards does not equate to specifying curricular content and
instructional environments. The supporting materials provided by the BC Ministry of Education
may still be used in transmissionist teaching methods; in fact, the first suggested use of the
performance standards is for classroom and standardized tests (BC Performance
Standards,2002). However, this is contradictory to the disclaimer that “relatively short questions
with one correct procedure and answer are not appropriate for performance assessment” (BC
Performance Standards,2002, p.220). The Performance Standards set goals, but they do not
present the means to achieve them. While there is a shift in pedagogical theory that learning
should be active, engaging and student-centred, the BC Ministry of Education provides little
support to teachers to make the shift in their classroom practices.
Teachers may benefit from direct suggestions of how to implement Performance Standards in the
mathematics classroom and still meet the curriculum requirements. This paper will investigate
transmission teaching and learning practices in the mathematics classroom and contrast it with
constructivist theories of teaching and learning. The use of constructivist teaching methods
through writing in the mathematics classroom will be explored. Finally, teachers will be
presented with tangible ways to implement writing in the mathematics classroom.
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Research Methods
A literature review will be conducted to gather information on the topics of transmission method
mathematics teaching, constructivist mathematics teaching, and writing in the mathematics
classroom. Predominately journal articles within the last 20 years will be accessed.
Definitions
Numeracy: the application of mathematical skills in daily activities. Numeracy involves the
ability of an individual to choose and apply appropriate mathematical procedures to solve
problems (BC Performance Standards, 2002).
Transmission method: explicit manner of teaching in which the teacher is the conduit of
information that is impressed upon the students; specific methods are deemed correct and
acceptable by the teacher (Aspinwall & Miller, 1997; Cobb, 1988).
Constructivist theory: essentially a philosophy about learning and teaching in which knowledge
is actively constructed by the learner (Pugalee, 2001).
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Limitations and Delimitations
There are many methods of possibly improving mathematics teaching, learning, and assessment.
This paper will focus on the task of writing in the mathematics classroom as well as the impact of
writing on students with learning difficulties. Specific learning disabilities, such as dyslexia or
dyscalculia, will not be addressed due to the word limit placed on this paper. Students with
learning disabilities in literacy but with strong mathematics skills will also not be addressed in
this paper.
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CHAPTER 2
Literature Review
Transmission Theory
Mathematics has been traditionally taught using the memorization of rules and formulae to reveal
a correct answer (Miller, 1991). Learning was obtaining parcels of information passed from
teacher to student; information would be retrieved exactly as transmitted for performance on
exams (Aspinwall & Miller, 1997). Burns (2003) posits that the primary goal of elementary
mathematics is to develop student competency of basic arithmetic skills and that problem solving,
justifying, reasoning, estimating, and reflecting are absent from most classrooms. Students who
achieve good grades have the ability to quote rules and plug numbers into equations (Miller,
1991). Gurganus (2007) describes the learner in a transmission method classroom as a “passive
recipient of irrelevant information transmitted by an uninvolved teacher” (p.50). Furthermore,
students themselves perceive mathematics as “learning how to solve problems in which there is
one right method and one right answer” (Burns, 2003, p. 6). This transmission model of
education in which teachers present problems and then show students how to solve the given
problems using already-proven theories has also been labelled ‘teaching by imposition’ (Cobb,
1988). This manner of teaching focuses on basic facts, isolated goals, direct instruction, and
pencil and paper exercises (Cobb, 1988; Countryman, 1992). Cobb (1988) argues that students
are constrained by the teacher’s explicitly taught methods. Failure of a student to understand a
concept would result in a teacher repeating the instruction or providing more worksheets (Cobb,
1988). However, many students will “sit and stare, not ask for help, and likely not fully
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understand” (Venne, 1989, p.66). Teachers do not realize the complex level of abstract
mathematical ideas they are presenting and may often overestimate the student’s understanding
(Miller & Mitchell, 1994). In this way, transmissionist instruction fails to engage and support
students in the mathematics classroom. Furthermore, Brownell and Moser (1949) assert that
“learning not grounded in understanding is pseudolearning…and is not worth the time and effort
required to achieve it” (as cited in Silver, 1999, p.155). The concentration on content and process
does not produce numerate individuals who can function outside of the classroom.
Transmission Theory and Students with Learning Difficulties
With the movement towards inclusive education, teachers are faced with a classroom of diverse
learners (Countryman, 1993). Students with learning difficulties “struggle with understanding
what mathematics means” and the transmission method of teaching mathematics does not
enhance their learning (Miller & Hudson, 2006, p28). The slowness or inability that students
with learning difficulties have when mastering basic skills may prevent them from engaging in
the entire task (Goldman & Hasselbring, 1997). They may memorize basic facts and procedures
without understanding concepts; furthermore, students with learning difficulties spend a
disproportionate amount of time in low-level practice and seatwork, thus not allowing them to
apply or understand the content (Baxter, Woodward, & Olson, 2005; Goldman & Hasselbring,
1997). These students also receive inferior instruction and, if placed in a lower level class, are
rarely exposed to rigorous academic content, critical thinking, or problem solving (Pugalee,
2001). Students with learning difficulties are therefore not expected nor challenged to become
numerate individuals.
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Current Theoretical Impetus: Constructivist Theory
Numerous investigations have revealed that isolated skill instruction fails to enhance students’
procedural and conceptual knowledge and application thereof (Goldman & Hasselbring, 1997).
As a result, a major shift in pedagogy of mathematics is evidenced by new learning standards set
by both the British Columbia Ministry of Education (BC Performance Standards, 2002) as well
as the National Council of Teachers of Mathematics (NCTM, 2006). The focus moves from the
product of mathematics to the process of mathematics, from the “memorization of isolated facts
and procedures to … conceptual understandings, multiple representations and connections,
mathematical modeling, and mathematical problem solving (NCTM, 2006). Goldman and
Hasselbring (1997) reason this shift to constructivist theory is as a result of the emphasis on the
need for all students, especially those with learning difficulties, to learn to solve problems, to
learn to reason, and to learn how to learn on their own. Miller (1991) also notes the movement
from correctness and product to process, content and understanding; it is no longer acceptable for
students to be able to merely manipulate symbols they do not understand nor correctly apply.
The constructivist theory embraces the teaching of whole concepts, the construction of
knowledge by students to solidify their knowledge, and the focus of mathematics on applications
rather than basic facts. The student’s construction of knowledge is shaped by communication
with others and the environment as well as his or her own metacognitive skills. Gurganus (2007)
argues that constructivism is a philosophy, rather than a single theory, as there seems to be little
agreement as to the definition and classroom application. He describes opposite ends of the
constructivist-teaching spectrum: the endogenous constructivist and the exogenous
constructivist. The former envisions the teacher as the “facilitator of holistic, self-regulated
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learning” as the student is fully responsible for his or her own learning (Gurganus, 2007, p.49).
The latter assumes that the teacher is essential to learning and that instruction must be direct,
controlled, and explicit. Some theorists, however, argue that there must be a balance between
direct instruction and student-lead explorative learning (Goldman & Hasselbring, 1997).
Gurganus (2007) describes this more balanced perspective being taken by the centralist
constructivist. The centralist constructivist may agree with the basic tenants underlying the
philosophy of constructivism: all knowledge is constructed by the learner; children are naturally
active learners; the gaining of knowledge and understanding occurs with full participation;
curriculum should be relevant and student-centred (Gurganus, 2007). These teachers may take an
active, but not central, role in student learning by scaffolding student knowledge, focussing on
metacognition and higher-order thinking, and promoting risk-taking, exploration, and
collaboration (Gurganus, 2007).
The desired result of this pedagogical shift is to raise the standard of higher-order thinking
(Goldman, 2002; Pugalee, 2001). Students are expected to reason, analyse, and interpret; they
should be able to “estimate the validity of their answer” and not rely on teachers or answer keys
for correct solutions (Cobb, 1988, p.99). Their learning should actively build connections
through the meaning-making process. The movement to constructivist methods of teaching
stresses the importance of learning in meaningful contexts (Countryman, 1993; Goldman &
Hasselbring, 1997). The current educational standards imply that students should be immersed in
active learning environments in which they are constructing their own knowledge of
mathematical concepts and connecting it to their existing web of information. In this manner,
students will have stronger links between procedural and conceptual knowledge and be more able
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to select appropriate procedures when faced with a problem as they have a deeper understanding
of the concept (Goldman & Hasselbring, 1997).
Constructivist Theory and Students with Learning Difficulties
It is argued that to learn mathematics, students must construct it for themselves (Countryman,
1991). This is also true for students with learning difficulties. In the past, the primary goal of
mathematics for students with learning difficulties was basic mathematical literacy through the
acquisition of basic mathematical skills (Goldman & Hasselbring, 1997). However, now that the
standards for math literacy have been clearly defined as the ability to apply mathematical
concepts, the standards for teaching students with learning difficulties should also evolve.
Students with learning difficulties should not be excluded from the opportunity to develop
mathematical life skills and the opportunity to gain conceptual knowledge to solve problems. .
In fact, students with learning difficulties often demonstrate conceptual performance that exceeds
what would be predicted based on their current performance level (Graham & Harris, 1989, as
cited in Goldman & Hasselbring, 1997). Constructivist principles can support underachieving
students in their study of mathematics by embedding important skill learning in meaningful
context (Pugalee, 2001). Students with learning difficulties may therefore be the students who
most benefit from constructivist teaching methods.
However, how does one ascertain that students have achieved a deeper understanding of a
concept? That their constructed knowledge is indeed based on known mathematical truths? One
can “never know with absolute certainty what is going on inside each students’ head” (Cobb,
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1988, p.92). Teachers who employ constructivist methods must therefore have methods to assess
students’ understanding. Writing in the mathematics classroom is a tool that does just that.
Constructivism Through Writing
The implementation of a constructivist approach to mathematics, specifically through writing,
ideally assists students to effectively refine and communicate their mathematical knowledge
(Aspinwall & Miller, 1997; Baxter, Woodward, Voorhies, & Wong, 2002; Burns & Silbey, 2001;
Jurdak & Zein, 1998). The improved ability of students to communicate mathematically, through
discussion and writing, is a goal of the BC Ministry of Education (1995, as cited in Leidtke &
Sales, 2001). As well, the NCTM (1998, as cited in Pugalee, 1999) identifies communicating
mathematically as one of the five processes through which students obtain and use their
mathematical knowledge. The ability of a student to communicate his or her understanding helps
solidify his or her connections between past knowledge and new knowledge (Miller & Hudson,
2006). Emig (1977) posits that writing is a “unique mode of learning” in that it provides
information processing at three different levels- motor (the physical act of writing), sensory
(reading what is being written), and cognitive (thinking about what is being written) (as cited in
Aspinwall & Miller, 1997). Writing seems to help make students internalize knowledge.
Pugalee (1999) argues that writing “creates an environment that supports the type of
metacognitive thinking that, in turn, supports mathematical reasoning” (p.21). The opportunity
for students to clarify, refine and consolidate thinking allows them to learn the complexities of
mathematics, to practice using mathematical vocabulary, and to articulate abstract concepts in
their own language. It also provides opportunities for students to make their own connections
between ideas and processes. This mathematical reasoning and metacognition helps students
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understand why they are using a specific procedure and they will therefore be better able to
transfer this knowledge to other situations (Goldman & Hasselbring, 1997). Students who reason
at a higher cognitive level tend to use a larger repertoire of thinking styles than other students;
they tend to take a deeper approach to learning (Zhang, 2002, as cited in Goldman, 2002).
Writing helps to develop metacognitive thinking, thus enhancing learning.
The Theory of Writing
Writing as a tool for learning in which ideas, connections, and concepts (instead of grammar and
punctuation) are assessed has been supported by many theorists (Vygotsky, 1962; Langer, 1970;
Britton, 1982, as cited in Durst & Newell, 1989). Students use the written word to communicate
ideas regarding a specific subject, thus allowing teachers insight into what the student actually
understands. For students to be able to fully understand a mathematical procedure, they must be
able to think, talk, and write about it (Van de Walle, 1995, as cited in Liedtke & Sales, 2001).
The NCTM (2006) recommends that students should be able to communicate mathematical
concepts by “speaking, writing, demonstrating, and depicting them visually” (as cited in
Williams & Wynne, 2000, p. 132). The BC Ministry of Education proposes that writing and
communicating mathematical concepts is an integral component of mathematical literacy (BC
Performance Standards, 2002). However, communicating mathematical concepts may pose
challenges for students with learning difficulties. Firstly, students with poor academic
performance may be marginalized by their teachers and by their peers, making them less likely to
participate in whole-class discussions (Pugalee, 2001). These students typically remain silent as
classroom conversations are dominated by capable and highly verbal students (Baxter,
Woodward, Voorhies & Wong, 2002). Even in smaller groups, students with learning difficulties
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remain passive and have marginal discussion (Baxter, Woodward & Olson, 2005). In addition,
conceptual and verbal demands, as well as the vocabulary, may be overwhelming to students with
learning difficulties. How then, can all students, especially those with learning difficulties, be
included in communication of mathematical concepts?
Allowing students to write about their understanding of mathematical concepts invites all
students, including those with learning difficulties, into the mathematical conversation. There is
a need for students to be able to communicate mathematical thinking- it allows them to engage in
construction of mathematical knowledge (Baxter, Woodward, & Olson, 2005). It also allows
them to reach non-cognitive goals of instruction such as improved attitudes towards mathematics
and improved metacognition (Cobb, 1988). Through this communication students make sense of
mathematics, thereby increasing their confidence as math learners. Students are able to articulate
their understanding of complex or abstract concepts as well as their misunderstandings. While
the transmission method of teaching mathematics focuses on the breakdown of communication
(where the student went wrong), constructivism focuses on what the student already knows and
builds from there. The focus is on successful communication (Cobb, 1988). Students are able to
practice using proper vocabulary, estimate and justify their answers, describe their methods, and
predict outcomes. Writing in the mathematics classroom helps develop a deeper understanding
of mathematical concepts in contexts to which students can relate. Writing is linked to improved
learning of mathematical content and improved ability of skills (Stonewater, 2002). While
writing is a very useful communication tool to explore mathematical concepts, studies have
shown that it is beneficial both to students and to teachers.
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Advantages and Drawbacks to Students
Students may be hesitant to begin to write in the mathematics classroom. The process of writing
brings to the surface the level of understanding students have of a mathematical concept. This
can be a daunting task even for college calculus students: “I’ve always had problems explaining
to others. It’s very frustrating” (anonymous student, as cited in Aspinwall & Miller, 1997, p.8).
Quinn and Wilson (1997) found that “writing seems to cause students to become confused and
frustrated and want to quit” (p.18). Students may be reluctant to write in the mathematics
classroom as the task forces them to disclose their lack of understanding or reveal the
inconsistencies in their reasoning; rote responses do neither and that may be why students feel
more comfortable with them. Perhaps if students were introduced to writing in the mathematics
classroom at an earlier stage in their education, there would be less resistance and frustration.
However, if teachers continue to encourage students to write about mathematics, they may find
that it benefits student conceptual understanding (Aspinwall & Miller, 1997). As students’
confidence increases in writing, their logic becomes more clear and focused and they may come
realize the importance of being able to communicate their knowledge effectively (Mayer &
Hillman, 1996). Teachers may even find that writing tasks are useful in classes with ESL
students if they are able to write in their first language (Quinn & Wilson, 1997). The resistance
from students and the initial poor quality of writing may deter teachers from continuing to write
in the mathematics classroom; however, the benefits provided to students outweigh the cries of
protest. Indeed, if teachers were to succumb to the students’ resistance to writing, this would
merely allow the continuation of misunderstandings and may reduce the likelihood of students
grasping the concepts of mathematics.
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The benefits to students of writing in the mathematics classroom are numerous (Aspinwall &
Miller, 1997; Burns, 2003; Hansen, 2000; Jurdak & Zein, 1998; Mayer & Hillman, 1996; Miller,
1991; Quinn & Wilson, 1997; Stonewater, 2002). Britton (1982, as cited in Durst & Newell,
1989) supports the use of writing to explore, organize, and refine ideas. Writing helps enhance
mathematical understanding, develops metacognitive skills, and improves students’ ability to
communicate mathematical ideas. Furthermore, writing has positive affective benefits as well.
Conceptual Understanding
Baxter, Woodward, and Olson (2005) describe the initial phases of the levels of writing as
recording and summarizing. In these two phases, students record information as directed by the
teacher and describe concrete examples in their own words. The focus is on facts, definitions,
procedures, and summaries. Simply by doing this, students are practicing using proper
mathematical vocabulary as well as communicating their level of understanding of procedures
(Hansen, 2000; Stonewater, 2002). Students must think about and use various symbols to
represent their thoughts in a mathematical manner (Fuqua, 1998). Furthermore, Quinn and
Wilson (1997) discovered that students in the third grade who participated in expository writing
in the mathematics classroom displayed an improved ability to retain details and concepts. As
students advance to the intermediate or more complex phases of writing, they are able to apply
and clarify the concepts; this analytic writing develops a stronger representation of content as
students embed it within their own constructed context (Baxter, Woodward, & Olson; 2005;
Durst & Newell, 1989; Pugalee, 2001; Stonewater, 2002).
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Writing develops thought processes that improve mathematical understanding, as students are
required to write clearly about a concept. If they are successful in writing clearly, they probably
understand the concept (Johnson, 1983, as cited in Miller, 1991). Quinn and Wilson (1997)
observed that secondary students gained a better conceptual understanding and improved their
understanding of algebra by writing their own problems, sharing ideas, and summarizing
concepts. In college level mathematics, one third of students declared that journal writing helped
improve their mathematical knowledge and problem solving skills (Borasi & Rose, 1989, as cited
in Stonewater, 2002). One student noted, “With writing, you are forced to display a conceptual
understanding of what you’re doing” (anonymous student, as cited in Aspinwall & Miller, 1997).
When students are asked to use writing to describe their mathematical understanding, they are
able to develop a deeper and more complex understanding and retention of concepts by being
able to explain their reasoning and defend their solutions (Aspinwall & Miller, 1997; Burns,
2003; Durst & Newell, 1989). This is evidenced in studies by Thaiss (1982, as cited in
Stonewater, 2002) in which students who kept journals scored 15% higher on tests. They also
showed increased ability in problem solving. Venne (1989) posits that once students have
written word problems, they can understand them and therefore solve them more effectively. In
summary, one teacher who incorporated writing in the mathematics classroom articulated that
there was “an emerging attitude by students that writing about their mathematical understanding
is beneficial as a strategy or process for learning” (Aspinwall & Miller, 1997, p.5).
Metacognitive Benefits
The use of writing in the mathematics classroom also enhances the metacognitive skills of
students. As students clarify and revise their writing, they are essentially clarifying their
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understanding as well as thinking more deeply about their thinking (Burns, 2003; Miller, 1991).
In a study by Borasi and Rose (1989, as cited in Stonewater, 2002), one half of students who
were writing in the mathematics classroom said, as a result, they were more aware of how they
learn math. Writing requires students to think, focus, and internalize concepts in a self-initiated
manner (Jurdak & Zein, 1998). Students come to realize multiple ways to solve problems and are
given the opportunity to investigate those ways (Hansen, 2000). Writing focuses on what
students are thinking and how they arrive at their solutions. By helping to develop higher-level
thinking skills, writing invites students to become more involved in the learning process (LeGere,
1991, as cited in Quinn & Wilson, 1997).
Communication Benefits
As improved communication of mathematical concepts is a major goal of the NCTM and the BC
Ministry of Education, it is important that there are suggested methods for teachers to help reach
that goal. Writing in the mathematics classroom helps students improve mathematical
communication, as they are able to practice using mathematical vocabulary. The written work
initiates dialogue between teacher and student or between student and student; weak arguments
are challenged, thinking is questioned, and clarity is required (Burns, 2003). This is of
significant importance for students with learning difficulties as they may not ask for assistance or
have the ability to articulate what they do not understand (Miller, 1991). More specifically,
students with learning difficulties participate less in class discussions but are more motivated to
communicate through writing (Miller, 1991; Baxter, Woodward, & Olson, 2005). Writing
provides a non-threatening environment for students to express their understanding.
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Not only does writing in the mathematics classroom improve writing skills, students are also
more able to verbally communicate their understanding. Just as teachers use words to teach and
textbooks use written words to convey concepts, students can be provided the opportunity to
verbalize their own mathematical understanding (Venne, 1989). Students are encouraged to
justify and explain a solution in writing and are therefore more able to verbally discuss solutions
after having written about them (Baxter, Woodward, & Olson, 2005; Quinn & Wilson, 1997).
Again, this is important for students with learning difficulties as they often withdraw from whole-
class discussion, especially when unprepared (Baxter, Woodward, & Olson, 2005). Writing
allows them to practice using vocabulary and clarify their thinking before verbal discussions
occur. The time given to formulate their response and articulate their questions may aid them in
becoming more active participants in the classroom. Writing in the mathematics classroom
enhances students’ ability to communicate, both on paper and verbally, their understanding of
mathematical concepts. Students themselves come to realize the importance of being able to
communicate their knowledge effectively (Mayer & Hillman, 1996).
Affective Benefits
Traditionally, math was often taught symbolically with one method used to achieve one right
answer (Miller & Mitchell, 1994). This transmission method of teaching, coupled with
traditional assessment methods of paper and pencil tests, lead to the development of mathematics
anxiety in some individuals (Miller & Mitchell, 1994). Many people react so negatively to
mathematics “that their ability to memorize, concentrate, and pay attention is effectively
inhibited” (Kogelman & Warren, 1978, as cited in Miller & Mitchell, 1994). This state of panic
and paralysis prevents some students from learning mathematical concepts effectively. However,
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Morris (1981, as cited in Miller & Mitchell, 1994) found that anxiety does not correlate with
intelligence but that there is a strong inverse correlation between the degree of mathematics
anxiety and persistence. Baxter, Woodward, & Olson (2005) found that there is a strong
relationship between interest and achievement in mathematics and that “mathematical proficiency
includes a productive disposition” (p.131). Therefore, a student’s attitude toward mathematics
may be a more important factor than previously thought. Unfortunately, students with high levels
of mathematics anxiety avoid math both in schooling and career choices; when they do take a
mathematics course, they receive lower grades even though they may be competent (Ashcraft,
2002).
Research has found that writing in the mathematics classroom helps dissipate feelings of anxiety
(Aspinwall & Miller, 1997; Miller & Mitchell, 1994). Even more importantly, writing can
develop positive attitudes towards mathematics (Aspinwall & Miller, 1997; Jurdak & Zein, 1998;
Mayer & Hillman, 1996; Miller, 1991; Stonewater, 2002). Journal writing provides students with
opportunities to discuss their frustrations or misunderstandings and to seek assistance privately
(Miller & Mitchell, 1994). As noted above, writing helps promotes student academic
understanding of mathematical concepts thereby empowering students as self-determined learners
(Ryan, Rillero, Cleland, & Zambo, 1996). Furthermore, Ashcraft (2002) hypothesizes that the
anxiety an individual experiences with mathematics is due to the disruption of the working
memory of the brain as the individual focuses on the anxiety, not on the mathematical question.
Writing could alleviate some of these stressors, as rote answers are not usually required, therefore
less stress is placed the working memory. Countryman (1993) posits writing helps students to see
themselves as agents of their own success and therefore as mathematicians. Because they are
able to communicate their ideas, their interest in and knowledge of the subject improves.
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Students also develop confidence to write about more complex topics (Baxter, Woodward, &
Olson, 2005). Writing in the mathematics classroom motivates students to become active
learners by decreasing anxieties and increasing interest in the subject.
Advantages and Drawbacks for Students with Learning Difficulties
Students with learning difficulties also benefit from writing in the mathematics classroom.
Morocco (2001) outlines the research of the REACH institute on teaching for understanding with
students with disabilities. The premise under investigation is that students with learning
difficulties will show improved understanding when they engage in four research-based
principles of teaching for understanding. The principles are: instruction is designed around
authentic tasks; there are opportunities to develop cognitive strategies; learning is socially
mediated; and there is engagement in constructive conversations. Writing in the mathematics
classroom is an activity that can incorporate these four important instructional strategies.
Authentic tasks should engage students in constructing their own knowledge and improve
understanding through the process of integrating prior knowledge with new information
(Morocco, 2001). Questioning, organizing, interpreting and synthesizing are all activities that
facilitate the construction of knowledge. With effective prompts provided by the teacher such as
those provided in Appendix A, writing in the mathematics classroom can incorporate these
activities. Finding alternate methods of arriving at a solution, comparing and contrasting
concepts, applying concepts to real-life situations, designing their own word problems, and
asking questions about misunderstandings are some of the ways in which students with learning
difficulties can start to construct their own mathematical knowledge through writing.
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Writing allows students with learning difficulties to develop cognitive strategies, as reflection
upon their own learning can be a writing topic in the mathematics classroom. Writing can relieve
the load on the working memory a student with learning difficulties’ as the focus shifts from fact
recall to conceptual understanding and personal reflection. Furthermore, students can use writing
to build upon their intuitive understanding of concepts and rely less on memorized facts. This is
extremely beneficial to students with learning difficulties, as there may be fact-recall difficulties
or working memory deficits (Goldman & Hasselbring, 1997).
Social interaction is a third essential role in the construction of knowledge for students with
learning difficulties (John-Steiner & Mahn, 1996, as cited in Morocco, 2001). Students need to
feel a sense of ownership and should be encouraged to “make their thinking visible to one
another through talk, [and] visual representations” (Morocco, 2001, p.8). Writing in the
mathematics classroom allows students with learning difficulties to formulate their responses
before sharing with classmates, thereby increasing their thinking time and perhaps alleviating
hesitancies or anxieties. It thereby encourages students with learning difficulties to participate in
classroom discussions, something that they may not do otherwise (Baxter, Woodward, & Olson,
2005). As further suggestion to improving social interaction, Morocco (2001) outlines the need
for students to work on problems that are “rich enough to invite a variety of perspectives” (p.8).
By utilizing a variety of mathematical writing prompts, students can be encouraged to write about
their own mathematical experiences and perspectives. Alternate solutions, attitudes towards
mathematics, and possible applications of concepts can all be prompts to support a variety of
perspectives. A list of possible prompts for Grade 8 Mathematics is included in Appendix A.
22
Constructive conversations improve understanding for students with learning difficulties as their
thinking is brought to the forefront (Morocco, 2001). A student with learning difficulties may
devote too much time trying to master a skill that may be irrelevant in the context of a larger task.
Their slowness or inability to learn an isolated skill prevents the student from engaging in a larger
task (Goldman & Hasselbring, 1997). They may have never been challenged to complete a task
as teachers may have assumed that they don’t have the isolated skills and therefore the ability to
complete the task. In this manner, students with learning difficulties are excluded from
constructive conversations in the classroom. Writing in the mathematics classroom invites
students with learning difficulties into the conversation as the focus shifts away from basic skill
acquisition. The conversation encourages students to connect, compare, and contrast their
understandings of concepts. They can express their questions and practice and internalize their
ways of thinking. Through the process of writing, students can participate in their own
discussions and then share their work with others. This sharing process provides opportunity to
assimilate others’ perspectives and understanding with their own, thus enriching their
understanding of the mathematical concepts.
Morocco (2001) specifically posits that one of the most beneficial teaching methods for achieving
the goals of understanding for students with learning difficulties is the use of assessment methods
that reveal the thinking processes of the students. Morocco (2001) highly recommends the use of
journal prompts to allow students with learning difficulties time to practice thinking processes
essential to success. The practice of journal writing in the mathematics classroom assists teachers
in planning instructional support for students with learning difficulties, ultimately benefiting the
student.
23
Advantages and Drawbacks for Teachers
The NCTM (2006) encourages all teachers to incorporate writing in the mathematics classroom
to develop deeper understanding of concepts and principles. If the NCTM recommends writing
in the classroom, it is interesting and troubling to note that Silver (1999) recorded that a majority
of the mathematics teachers surveyed were either unfamiliar with the use of writing in the
mathematics classroom or never or rarely implemented it. Applebee (1981) discovered that
although teachers believe writing has a place in all subject areas, there is widespread confusion as
to what benefit writing may have towards learning. What is preventing teachers from utilizing
writing strategies to improve students’ mathematical understanding? Goldman (2002) argues that
teachers teach the way in which they have been taught. As instruction in the mathematics
classroom has traditionally focussed on basic facts through the use of worksheets and drill and
practice, teachers find difficulty in adapting new ideas into their existing teaching repertoire
(Goldman, 2002). Additionally, teachers may have concerns that writing in the mathematics
classroom may allow students to stray from the “rigor and precision of formal mathematics” by
encouraging students to construct their own knowledge (Baxter, Woodward, & Olson, 2005,
p.132). External pressures may also be preventing teachers from exploring the use of writing in
the mathematics classroom. Teachers feel constrained by the prescribed learning outcomes set by
external governing bodies (the BC Ministry of Education, for example) as well as the pressure of
standardized testing to incorporate constructivist teaching methods in the mathematics classroom.
Teachers may feel the pressure, through external exams as well as the Prescribed Learning
Outcomes, to cover the material (Aspinwall & Miller, 1997, Quinn & Wilson, 1997). There is a
generalized concern that class time is limited and the teacher must find the most effective or
efficient means to cover the curriculum (Aspinwall & Miller, 1997; Fuqua, 1998; Quinn &
24
Wilson, 1997). Writing may be viewed as a time-consuming approach as students need time to
write, teachers need time to respond and to continue the mathematics conversation. Teachers
may feel that the transmission method is a more efficient way of disseminating information as the
lecture is given, students practice the process, and the quiz is administered (Aspinwall & Miller,
1997). A teacher, however, will be more willing to implement writing if students are
academically above-average, negating the argument that writing is too time-consuming. Lastly,
teachers argue that students have difficulty writing therefore having them write about
mathematical concepts is too daunting a task. One teacher noted, “writing seems to cause
students to become confused, frustrated, and want to quit” (Quinn & Wilson, 1997, p.18).
Students even express their annoyance of the task: “I’ve always had problems explaining to
others. It’s very frustrating” (Aspinwall & Miller, 1997). Initially, writing may be of poor
quality, vague, and shallow possibly reflecting a lack of effort, as well as a lack of writing ability,
of the students (Stonewater, 2002). They will avoid complex concepts and the writing may be
difficult to understand (Stonewater, 2002). Teachers may become discouraged by poor results
and thus abandon writing in favour of transmission instruction.
As mentioned earlier, poor student response in writing due to either reluctance to participate, low
effort, poor quality may be largely due to the student’s reluctance to reveal his or her lack of
understanding. Teachers failing to recognize this may be discouraged from continuing with
writing experiences on the grounds, ‘the students don’t like it and don’t do it’. The intent of this
paper is to demonstrate to teachers that the students’ reluctance to write in the mathematics
classroom is the strongest argument for the need for them to do just that- write! Simply put, if we
want to change our levels of mathematics literacy, we must change our teaching techniques.
25
Including All Students
Most teachers have a favourable attitude towards the use of writing in the mathematics classroom
and those who have implemented writing can speak of the benefits (Quinn & Wilson, 1997). By
using prompts for writing activities at the beginning of class, teachers can quickly have students
thinking mathematically and trying to make connections (Miller, 1992). Students are therefore
on task and making the transition to become active learners. The act of writing allows students to
have success as it challenges some students to make complex connections and deductions and it
provides opportunity for struggling students to explain their logic and reasoning as well as seek
clarification. As the focus is on the process rather than the product, students of all capabilities
can feel like successful mathematical communicators. In this manner, teachers are able to make
connections with every student and provide each student with the opportunity to demonstrate
understanding (Mayer & Hillman, 1996).
Lesson Planning
As students write about their understanding of mathematical concepts, teachers gain valuable
insight into what students know and don’t know (Countryman, 1993). The writing also informs
the teacher of the students’ construction of knowledge; any errors or misconceptions in the
construction of this knowledge can be identified and quickly corrected (Aspinwall & Miller,
1997). Teachers can identify student strengths to build upon as well as any areas that may need
remediation (Jurdak & Zein, 1998). From this, a teacher is able to reflect upon previous lessons
and plan more effective lessons based on what students know and what they don’t know. This
important form of assessment for learning allows teachers to teach to students’ knowledge base,
26
instead of to curriculum. The pace and content can be driven by student knowledge instead of a
teacher’s perception of what has been covered and what needs to be covered (Miller, 1991).
A Shift in Pedagogy from Assessment of Learning to Assessment For Learning
As there is a shift in pedagogy, there follows a shift in assessment. In the transmissionist
mathematics classroom, assessment is generally through quizzes and paper and pencil tests. The
teacher collects a wealth of quantitative information to report on the progress of his or her
students. The students can demonstrate their understanding by correctly solving problems using
the anticipated method. These summative assessment methods are also used by the BC Ministry
of Education in the forms of Provincial Exams and Foundation Skills Assessment (FSA).
Provincial Exams are administered four times per year for students in grades 10 and 12 academic
subjects and are worth 20 and 40 per cent of the student’s final course grade. FSA is
administered each year to students in grades four and seven and provides an overview of student
learning in reading comprehension, writing, and numeracy as outlined in the Prescribed Learning
Outcomes as well as the Performance Standards (Foundation Skills Assessment, n.d.). The
province, school districts, and schools can use the results to evaluate how successful students are
at achieving basic skills and plan to improve achievement. Thus, the dichotomy exists- the BC
Ministry of Education encourages teachers to foster the student’s individual construction of
knowledge yet reinforces transmissionist teaching methods by staging rigorous mandatory
summative assessment.
27
Writing As A Tool For Assessment For Learning in a Constructivist Classroom
While the use of pencil and paper tests for summative assessment is fast, requires little
preparation, is easy to mark, parallels ministerial standardized testing, and usually receives
minimal dissent from students, it does not provide useful information that can be used in the
teaching process. In fact, overuse of assessment of learning, also called summative assessment,
in classrooms is detrimental to student learning (Black & Wiliam, 1998). In their metastudy on
assessment methods, Black and Wiliam (1998) clearly outline the negative impact summative
assessment has. The pencil and paper tests promote “rote and superficial learning” (p.141).
Quantity and presentation are assessed instead of quality of understanding. Assessment of this
nature is used to compare students, thereby downplaying individual improvement and learning.
Marks and grades are overemphasized and suggestions, strategies and metacognition are
underemphasized. In fact, research has discovered that marks given to students do not benefit
student learning; furthermore, assessment with marks and comments are also not beneficial as
students still focus on the grade (Black & Wiliam, 1998). It is documented that improved student
learning occurs only when comments are given, no grade (Black & Wiliam, 1998). These
findings may have significant impact on the methods in which teachers assess students. Lastly,
although teachers may be able to predict their students’ results on provincial standardized tests, as
their own classroom tests are so similar, they know little about the needs of their students. By
issuing pencil and paper tests and supporting standardized testing as forms of assessment of
learning, teachers are reinforcing rote learning and transmission methods of instruction rather
than focussing on the assessment of student work to discern learning needs.
There is, however, a place for provincial testing. Results hold school districts, schools, and
individual teachers accountable to the public education system. Teachers are required to teach
28
the mandated Prescribed Learning Outcomes and provincial testing ensures this. Testing also
provides a large-scale survey of student achievement over the years thus allowing governing
bodies to plan for educational development. Additionally, individual teachers must have a place
for summative assessment in their classrooms. They are responsible to report to students and
parents the final level of achievement of each student (Black & Wiliam, 1998). However,
summative assessment should not be the only form of assessment in the classroom as it only
provides a final achievement level, not a clear record of student understanding. Summative
assessments do little to promote education and their overuse is detrimental (Black & Wiliam,
1998). If summative assessments are to be used due to their ease, the use should be limited. As
educators, we should find other tools to provide for accountability- tools that actually measure
what we are trying to achieve.
Burns (2003) argues there are three main purposes of assessment: to provide the teacher with
information to change or revise instruction; to gain insight about student understanding; and to
provide a means with which to report to parents. Summative assessment fulfills only the last
purpose- formal reporting. While summative assessment may be the final assessment of what a
student has learned, formative assessment is continuous assessment of student progress that can
be used to restructure instructional delivery. In this manner, formative assessment fulfills all
three of the purposes of assessment as outlined by Burns (2003). It provides teachers with
information to help shape instruction, provides insight about student learning and metacognition,
and provides a means to report to parents. Burns (2003) posits that assessment is most useful
when integrated into classroom learning and instructional programming- formative assessment is
embedded within the classroom activities. Teachers provide continuous feedback to students on
projects, assignments and quizzes. If the feedback outlines the student’s strengths and
29
weaknesses, without any grade, the student’s learning will improve (Black & Wiliam, 1998).
The comments offered to students on tests, quizzes, assignments and projects should focus on
what each student can do to improve; furthermore, students must be given the opportunity and the
assistance to make the improvements (Black & Wiliam, 1998). The mathematical journal
provides opportunity for students to communicate their understanding as well as provides
teachers a medium in which to respond. Countryman (1992) articulates that reading math
journals tells her “considerably more about what students grasp and do not understand, like and
dislike, care about and reject” (p.28). The teacher is more aware of what students know and how
they have come to construct that knowledge.
The difficulty teachers face is finding the balance between assessing ongoing work to improve
student progress and understanding and fulfilling their roles as reporters of student achievement.
The pressure placed upon teachers by external forces, such as provincial exams, to have their
students perform to a certain standard may impede the improvement of teaching practice and the
incorporation of assessment for learning, such as writing in the mathematics classroom.
30
CHAPTER 3
Implementing Writing in the Mathematics Classroom
Although teachers have a favourable attitude towards writing in the mathematics classroom, the
majority of teachers do not implement this practice (Quinn & Wilson, 1997). However, there is a
sizable minority (27 to 39 per cent) who do write regularly in the mathematics classroom (Silver,
1999). These teachers are utilizing a teaching method that will enhance their students’
understanding of mathematical concepts and have proven that any teacher can implement writing
activities in the math classroom.
There are a number of factors a teacher may wish to consider before starting writing in the
mathematics classroom. Williams and Wynne (2000) recommend choosing the strongest
academic class to begin writing activities. In this way, a teacher will see more student success
and feel more comfortable with the activities. The teacher should also pre-determine what to
write, how long students will write for, the format students will follow, as well as how students
will be assessed (William & Wynne, 2000). It is recommended that affective journal prompts be
used as initial writing activities as they are one of the least intimidating writing activities for both
student and teacher. Alternatively, summarizing an activity or lesson may be an easy start for
students (Burns & Silbey, 2001). Journals should not be taken home and entries should be
completed during class time; this way, journals cannot be misplaced, forgotten or even read by
others without student consent. Teachers may choose to allow five to ten minutes for students to
write about a given mathematics topic. A list of possible writing prompts for Grade 8
Mathematics is included in Appendix A. The teacher may also want to inform students that
31
content and process will be the focus of assessment instead of grammar, punctuation and spelling.
This will decrease the anxiety some students may feel about writing while encouraging all
students, regardless of writing ability, to actively engage in the activity (Jurdak & Zein, 1998;
Ross, 1998; Stonewater, 2002; Williams & Wynne, 2000). Stonewater (2002) also recommends
that samples of high quality and lower quality writing be provided to students. Students learn by
modelling and seeing high quality work may help them to write. Teachers may also wish to
model the first few responses to the class by gathering class input and formulating a clear written
response. In this manner, students can safely contribute to a response and practice writing before
starting individual writing activities.
When to Write
Teachers can start a mathematics lesson with a writing activity by having students summarize the
previous day’s lesson and comment on difficulties they encountered or insights they gained.
Writing at the beginning of a lesson can help students collect their thoughts or ask questions
before a new topic is introduced. This may improve whole-class discussion by allowing students
to organize their thoughts and draw upon previous knowledge (Baxter, Woodward, & Olson,
2005; Elliott, 1996). Furthermore, teachers can use the discussion to immediately correct any
misunderstandings students may communicate as well as provide rich student-driven context to
begin a lesson. Writing can be implemented in the middle of a lesson if students are confused.
This quiet time of reflection allows students to start with what they know and understand and to
formulate questions based on what they do not understand (Baxter, Woodward, & Olson, 2005).
It also assists the teacher in redirecting the lesson should the need arise due to students’
misunderstandings. Writing activities may also be employed at the conclusion of a lesson.
32
Students may write a summary of the lesson or their thoughts or feelings of the lesson. The
teacher may also challenge the students to make connections between concepts such as “How are
fractions and ratios similar? How are they different?” The flexibility of when in a lesson the
writing process can be implemented may be attractive to a teacher who is beginning to utilize this
activity.
What to Write
Journaling
There are a number of different writing activities teachers can use to implement writing in the
classroom. Countryman (1993) describes free-writing as short entries about anything on the
given topic in which students write their thoughts as they come and ask questions as they arise.
This form of journaling promotes a private conversation between the student and the teacher.
The student is able to express his or her feelings, reflections, prior knowledge, and mathematical
understandings (Mayer & Hillman, 1996). Personal writing such as this closely reflects inner
speech and the teacher is able to gain valuable insight into the thought processes of his or her
students (Countryman, 1993). Initially, it may be difficult for students to be able to write freely
without self-editing or censoring their thoughts, but the teacher may wish to assure them that
their journal will neither be criticized nor evaluated on the first draft (Countryman, 1992). As the
initial writings may come slowly for students, the teacher may provide journal prompts to initiate
student response. The prompts may be procedural, conceptual or affective (refer to Appendix A).
Procedural or process prompts focus on mathematical process skills such as “Given a linear
equation, how does one determine the slope of the line?” Students are able to discuss how a
33
particular question is solved and give alternate methods of solving it. While the focus of the
question is process in nature, the student is using language to explain his or her procedures. In
this manner, students are able to verbalize the process and therefore gain a higher level of
communication skills. Conceptual prompts do not focus on finding the correct answer but
encourage the student to explain the reasoning behind the mathematical process. For example,
students may be asked to compare area and perimeter and give examples. This type of question
requires students to have a sound understanding of perimeter and area and be able to articulate
the relationship between them. The third type of prompt, affective prompts, may be the least
intimidating for students to respond to, as the prompt draws upon personal experience. Students
are able to discuss their successes, their mistakes, and their attitudes towards mathematics. They
may be presented with a prompt such as “When I think about fractions, I feel…” As affective
prompts may be easier for students to respond to because they can write from personal
experience, teachers using journal writing in the mathematics classroom are encouraged to begin
with affective prompts (Countryman, 1993).
Story Problems
Students may develop and solve their own story problems as student-generated work improves
problem-solving skills (Pugalee, 2001). The teacher may provide the students with a graph and
the students write a story based on the graph. Students can create a mathematical problem and
then solve it; alternatively, the teacher may solve a problem incorrectly and the students are
encouraged to discuss the errors the teacher made. As teachers become more familiar with
writing in the mathematics classroom, writing topics are more easily created. As students
34
become more comfortable with writing, they are better able to express their thinking, thus
enhancing the teaching-learning interaction.
Assessment Methods
Writing in the mathematics classroom will produce volumes of written material that may be
daunting for a teacher to assess. By realizing that writing is a tool for learning as well as a tool
for assessment, teachers can optimize their assessment techniques while minimizing their
workload. Teachers must abandon the thought that they need to correct all entries for
punctuation, grammar, spelling, or conceptual errors. The purpose of students’ writing is to
allow the teacher access to the students’ thinking and to gain insight into their learning.
Countryman (1992) believes that the “journal gives a teacher access to that thinking, but not
license to take on the burden of fixing it all” (p.39). She recommends that, at the end of the year,
students look over their journals, number the pages and select ten or so of their favourite entries.
They can then make a table of contents including those favoured entries, write an introduction,
and write a conclusion. Countryman (1992) recommends evaluating the journals for frequency of
entries, length of entries and self-initiated topics. Teachers should also not feel obliged to give
individual comments on every entry to every student; however, when comments are made, they
should be specific in nature and address the student’s writing (Burns & Silbey, 2001). The
feedback should focus on the mathematics and encourage the student to revise if necessary. By
following Countryman’s suggestions, a teacher can easily incorporate journal writing into the
mathematics classroom without devoting excessive amounts of time to proofreading and
correcting.
35
CHAPTER 4
Conclusion
There has been a shift in the pedagogy of mathematics. Governing bodies such as the NCTM and
BC Ministry of Education are emphasizing the importance of helping students become numerate
individuals. Students are expected to be able to estimate, interpret, apply, justify, represent, and,
of course, communicate in a mathematical manner. The transmission method of learning in which
students are presented with the information by the teacher and are focussed on the exactness of
calculations and formula has been replaced. The educational direction focuses on raising the
standard of higher-order thinking as well as reaching other non-cognitive goals of education.
Students are expected to be independent learners who have the ability to construct their own
knowledge and build new knowledge upon existing understandings. More specifically, students
are expected to become numerate individuals with the ability to apply mathematical knowledge to
diverse situations.
It is desirable to have theories of instruction that are compatible with and support the goals of
mathematics education. While the new standards for mathematics education are set, the possible
means to achieve them are not necessarily communicated to educators. Teachers may be unsure
of new theories of instruction or teaching methods to assist them in supporting the new standards.
While teachers must remain autonomous professionals who take responsibility for their own
teaching practices, instead of simply becoming conduits of information who follow expert
direction, supportive instructional practices should be incorporated into the newly developed
standards for mathematics education. The standards set goals but the means to achieve them are
36
not presented. Improving numeracy skills of students through writing in the mathematics
classroom is only one method of reaching the goals.
To learn mathematics, students must construct it for themselves. Vygotsky (1962, as cited in
Pugalee, 1999) argues that conceptual thinking is the strong link between the development of
thought and the development of language. How are teachers able then to witness this conceptual
thinking, especially in the mathematics classroom? How are teachers able to discern if students
truly understand a concept when mathematics has traditionally focussed on exactness and
product? Writing in the mathematics classroom allows teachers limited access into the cognitive
processes of their students. Students are able to communicate their understandings of concepts,
ask questions, clarify their reasoning and explore applications. They are able to enhance their
metacognitive abilities as well as improve their mathematical confidence. Writing in the
mathematics classroom allows teachers insight into the thought processes of their students thus
assisting them in instructional planning. Writing in the mathematics classroom can be used as an
effective tool for assessment for learning; journaling allows students and teachers to have an
ongoing dialogue about mathematical concepts and learning.
A teacher should be willing to challenge his or her students, to provide opportunities for them to
excel, and to help instil in them a sense of confidence as learners. It is hoped that this confidence
in, as well as understanding of, their own cognition will enable them to accept new challenges
and new problems. Teachers embracing the new mathematics learning standards to improve the
numeracy of students may be looking for ways in which to meet this goal; implementing writing
in the mathematics classroom is one of the methods to do so. The research upon which this
37
paper is based strongly suggests that incorporating writing into mathematical instruction can
significantly improve numeracy in all students, including students with learning difficulties.
38
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APPENDIX A
Possible Writing Prompts For Mathematics 8
Procedural:
• Explain perimeter and how to calculate the perimeter of a regular polygon.
• Explain the distributive property to a friend who missed this class.
• Explain how to add two fractions. Use diagrams to help your explanation.
• What is an algebraic expression? When are they useful?
• How do you create a table of values for an algebraic expression?
• Explain how to subtract negative numbers to your grade 5 brother.
• Explain the Pythagorean Theorem and give an example of when it may be useful.
• How do you calculate the area of a triangle? How can you remember the ‘formula’
without memorizing it?
Conceptual:
• What would happen if fractions were added without a common denominator? What is
wrong with this?
• Explain to your best friend why 4a + 5b = 9ab is incorrect. Draw a picture if necessary.
• Why should we estimate our answers? Is estimating a useful skill?
• How can you determine if two fractions are equal? Can you think of more than one way?
• How does what we learned today apply in real life?
• Jim and Sue solved today’s problem different ways. Which method would you use and
why?
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• Sally solved today’s problem in this way- _______. What is wrong with her procedure?
What is a correct way to solve it?
• When is ‘guess and check’ a useful problem-solving method? When is it not?
• How are ratios and fractions related? How are they different?
Affective/ Reflective:
• What was difficult to understand today?
• What did you learn today?
• What questions do you have about yesterday’s homework? About today’s lesson?
• When I think of fractions, I feel…
• My favourite memory of my math career is…
• My mathitude (attitude about math) is… because…
• Should students be permitted to use calculators in class? Why or why not?
• Write a poem about fractions. Use the terms numerator, denominator, equivalent.
• What is easier to understand: ratios, fractions or percentages? Why?
• How did you prepare for today’s test? How do you feel you did?
• What could you do to improve your mark in this class?
• How would you describe your math experience in this class so far?
• What can I do to help you better understand today’s lesson?