1. Adrian Jones M.A.(Ed.) Candidate
Secondary school mathematics teachers’ views of manipulatives and their use in the classroom
Adrian Jones
M.A.(Ed.) Candidate
Before I begin I would like to thank you all for your guidance in helping me get to this point…
The title of my proposal and this presentation is Secondary school mathematics teachers’ views of manipulatives and their use in the classroom
I’m looking at a longer title to make it even more of a mouthful!

2. Mathematical Manipulatives Research Perspective
FOLLOW ME…
Motivation
Mathematical Manipulatives
Research Perspective
What’s Really Happening in the Classroom?
Challenges
Influence of Teachers’ Views on Practice
Conceptual Framework
Research Questions
Data Analysis dimensions
Timeline
I would like to follow this outline, starting with my own motivation for this research proposal.
I know that the personal significance of this research was something that both Richard and Jerome encouraged during the writing of the proposal. I have taken this to heart, and this is where I start this discussion.
Starting in this way also encourages a chronological narrative to my thinking… So, I would like to present more of my own ‘train of thought’ that led to the research question by discussing some of the ideas that has led to my own views and understanding.
I shall discuss some of the research that situates and contextualizes the problem that I wish to investigate.
I shall also describe a conceptual framework that I have adapted to visualize key areas of the problem context and then define my research question, and some of the research-derived dimensions that may kick-start the data analysis.

3. THE MOTIVATION
Introduced to mathematical manipulatives during my teacher education program in 2006.
Challenged my own ‘understanding’ of mathematics
Many “ah-ha” moments, new insights (larger mental repertoire?)
Opportunity to develop understanding of their use with pre-service teachers during Math Camp
Helped my children in their own understanding by…:
connecting manipulation of algebra tiles to algebraic operations (e.g.: collecting like terms, factoring and reforming quadratics, multiplying binomials…).
helping to make concrete and meaningful connections to abstract mathematical concepts
shifting the locus of understanding from the teacher to the student
So why aren’t my children’s (and other) teachers, especially in high-school mathematics, using them for their students’ (and potentially their own) benefit?
I want to gain a better understanding of…
why and how ‘experienced’ teachers use manipulatives in their classrooms
benefits of and challenges in their use in secondary-school mathematics
the educational research process (CIIM+)
Within the context of this research topic, my experience with the teacher education program gave me two new yet related perspectives…
The first was that I started to think of mathematics from a pedagogical viewpoint… in that I started to think about math from the perspective of “how” and “why” rather than from the more familiar vantage of “what”.
The second was that I began to “see” mathematics differently (and I mean see with a capital S. Through the introduction to manipulatives I began to make connections between concepts that up to that point had been largely isolated and not “visualized”…
And recognizing that I was adding to an existing knowledge base, I began to realize the potential that manipulatives may have in providing another way to develop or deepen the understanding of math concepts.
Math Camp was an opportunity to see first hand how people, many of whom were self-professed math “phobic”, were able to start making new sense and connections between what had previously been inaccessible and disconnected ideas.
Working with my children in their struggles with high-school math, and in particular algebra, further convinced me that manipulatives can be powerful tools for learning mathematics.
Neither children had used manipulatives in high school (to date) yet seemed to benefit from their use at home. The inevitable question for me was why my children’s teachers did not use them?

4. MATHEMATICAL MANIPULATIVES
Manipulatives can be considered as concrete representations. Other external representations include pictorial, language, experiential, symbolic. (as proposed by Lesh, Post, & Behr, 1987)
Manipulatives are “designed to represent explicitly and concretely mathematical ideas that are abstract” and a catalyst in building representations of math concepts
real-world objects (e.g.: coins, cube-a-links) used for a variety of concepts, or
specifically designed materials (e.g.: algebra tiles, fraction circles) used to teach particular mathematical concepts.
to be touched, moved about, rearranged, and handled by students (Moyer, 2001; Kennedy, 1986; Pimm, 1995)
The understanding and value of manipulatives comes from being used by students and teachers engaged in shared and meaningful practices that involve strong and multiple connections to underlying mathematical concepts (Meira, 1998; Moyer, 2001)
So, what are mathematical manipulatives?
For me, manipulatives are concrete representations; one facet of a body of external representations …
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They are objects used to represent something else, or objects that we use to “stand-in” for something else.
They can be ordinary and familiar objects – coins, building blocks, buttons, cookies etc. or they can also be specially designed to help teach particular concepts. Algebra tiles, fraction circles, base 10 blocks are examples.
Whatever we choose, the key to making them useful mathematical tools is what we make them represent, or “stand for”, connecting them carefully to the mathematical concepts, and the types of activities that we design…

5. RESEARCH PERSPECTIVE
Studies report declining use of manipulatives as students progress through school years resulting in little use in high-school mathematics classes
(e.g.: Howard, Perry & Lindsay, 1996; Suurtamm & Graves, 2007; Weiss, 1994)
Why the decline of manipulative use in high-school when:
The use of manipulatives is strongly recommended by current curricula for all students, across all grades (NCTM, 2000; OME, 2005).
Researchers find that long-term use of manipulatives:
increases mathematics academic achievement (Ernest, 1994)
improves attitudes towards math (Leinenbach & Raymond, 1996)
has value to students in all school grades (LeNoir, 1989)
The studies that I could find that investigated manipulative use in high-school were largely quantitative in nature – the three mentioned here are examples – showed similar declines in use in later grades.
Howard and colleagues looked at manipulatives in primary, intermediate and secondary schools in Australia
Weiss reports on the The 1993 National Survey of Science and Mathematics Education, in the USA
Chris Suurtamm and Barbara Graves study entitled Curriculum Implementation in Intermediate mathematics included a questionnaire of over 1,000 intermediate teachers.
While some studies offered suggestions for this decline, most focused on the mere presence of manipulatives and the self-reported use of them by teachers.
I was unable to find any studies that focused on how and why teachers were (or perhaps more importantly, were not) using them in the classroom.
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So why this decline given the promotion of the use of manipulatives in the curriculum, in many curricular resources (TIPS for example) and the researched benefits to all, not just struggling, students…?

6. THE REALITY
Questionnaire data from 1,096 Ontario Grade 7-10 teachers, CIIM Report, Suurtamm & Graves (2007)
How important is promoting the use of multiple representations of ideas?
Very
Important
Grade 7
60%
Grade 8
53%
Grade 9 Applied
50%
Grade 9 Academic
37%
Grade 10 Applied
47%
Grade 10 Academic
30%
These data were gathered as part of the CIIM program with which many of you are very familiar.
I use this data to draw attention to the declining use of manipulatives in the higher grades…
To begin, clearly teachers must see value in using manipulatives and importance in using multiple forms of representation to introduce and reinforce mathematical concepts…

7. THE REALITY
Questionnaire data from 1,096 Ontario Grade 7-10 teachers, CIIM Report, Suurtamm & Graves (2007)
How important is promoting the use of multiple representations of ideas?
In this class, how comfortable are you with using concrete materials to teach mathematics?
Very
Important
A Lot
Grade 7
60%
46%
Grade 8
53%
43%
Grade 9 Applied
50%
48%
Grade 9 Academic
37%
30%
Grade 10 Applied
47%
35%
Grade 10 Academic
27%
Similarly, teachers’ comfort level with the use of manipulatives for teaching and learning is a big factor in their use

8. In this class, how often do students use manipulatives?
THE REALITY
Questionnaire data from 1,096 Ontario Grade 7-10 teachers, CIIM Report, Suurtamm & Graves (2007)
How important is promoting the use of multiple representations of ideas?
In this class, how comfortable are you with using concrete materials to teach mathematics?
In this class, how often do students use manipulatives?
Very
Important
A Lot
Never
Use
Grade 7
60%
46% 1%
Grade 8
53%
43% 4%
Grade 9 Applied
50%
48%
12%
Grade 9 Academic
37%
30%
40%
Grade 10 Applied
47%
35%
21%
Grade 10 Academic
27%
Perhaps these two facets – importance of the use of manipulatives, and comfort level in using them – explain why 1 in 2 teachers in this study report that they never use manipulatives in grade 10 mathematics …
So inevitably this encourages further investigation…. WHAT do teachers think about manipulatives, and HOW and WHY do they use them in the classroom

9. THE CHALLENGES
Teachers face many challenges in their use of manipulatives in the classroom:
Perceived value (Sherin, Mendez, & Louis, 2004)
Math meaning behind their use (Moyer, 2001)
Uncertainty how to use them (Suurtamm & Graves, 2007)
Time and curriculum coverage pressures (Weiss, 1994)
Changes to class environment (Howard et al., 1996)
But they are worth facing because:
By demonstrating how to use the manipulatives as tools for better understanding, teachers open doors for many students who struggle with abstract symbols.
(Moyer & Jones, 2004, p. 29)
However, research on how and why teachers use of manipulatives within secondary school mathematic classrooms is lacking (Howard et al., 1996, Moyer, 2001)
We know that teachers face challenges in their use… and this list in by no means exhaustive…
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and, if the research is to be believed, these challenges are worth facing
So, this leads to my attempt to better understand the issues....

10. INFLUENCE OF TEACHERS’ VIEWS ON PRACTICE
Teachers’ teaching practice is grounded in their beliefs and understandings of mathematics and their perceptions as to how children learn (Thompson, 1984).
Teachers’ beliefs of mathematics in general, and specifically in the value of manipulatives, influence the way in which they are used (Sherin, Mendez, & Louis, 2004).
So, to examine views and practice…
If teachers’ characteristic patterns of behaviour are indeed a function of their views, beliefs and preferences … then any attempt to improve the quality of mathematics teaching must begin with an understanding of the conceptions held by teachers and how these relate to their instructional practice
(Thompson, 1984, p. 106)
In considering teachers practice, I turned to Alba and Patrick Thompson (both separately and together), Deborah Ball, Paul Ernest and others
They examined teacher’s knowledge and their conceptions of mathematics (considered as an amalgam of views, beliefs, preferences and attitudes) and helped to make the connection between the these conceptions and their teaching practice.
Citing Thompson, Cooney and others, Azita Manouchehri (1997) puts it nicely when she suggests that “teachers translate their knowledge of mathematics and pedagogy into practice through the filter of their beliefs” (p. 198)
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Teachers’ views (consciously or unconsciously held) about mathematics teaching and learning, what mathematics is and what constitutes mathematical knowledge influences their methods of instruction and delivery, their beliefs of the appropriate way for students to learn, and the choice of texts and activities - which also includes their use of manipulatives.
This - the enacted curriculum – is what happens in the classroom
Not surprisingly, then, I realized that in order to better understand how and why teachers use manipulatives as they do, I should start from a better understanding of teachers’ views and beliefs about mathematics and the role of manipulatives in its teaching and learning…

11. Teachers Views & Beliefs
CONCEPTUAL FRAMEWORK
Examine views and knowledge to better understand practice
Social Context of Teaching
Teacher Knowledge
Pedagogy, practice & subject knowledge
Pedagogical
Content
Student
Learning
Reflection
The catalyst
Subject
Content
Current Curriculum
Teacher
Practice
Multiple
Representation
Curriculum
Classroom
Concrete reps.
(Manipulatives)
Theories and research
Enacted
curriculum
Teachers Views & Beliefs
Teaching of
Mathematics
Theories of learning
Learning of
Mathematics
Yet, these ideas needed some structure.
I created a conceptual framework that has its roots in frameworks developed by Ernest and by Koehler & Grouws, later extended by van der Sandt.
It also provided a useful structure for the proposal itself.
This framework allowed me to visualize how the teacher’s practice is influenced by his/her beliefs and knowledge of mathematics and how the mathematics curriculum - with its own set of underpinning beliefs, pedagogical and subject knowledge – comes to be enacted in the classroom.
As Ball (1992) suggests, “teaching with manipulatives is not just a matter of pedagogical strategy and technique” (p. 47)…. Moyer’s (2001) study of concluded that with the use of manipulatives comes the mandate to change the delivery of instruction, which in turn requires changes to teachers’ knowledge and beliefs about manipulatives and their role in its mathematics teaching and learning.
So, I consider the research problem…
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… using manipulatives as the catalyst
Nature of
Mathematics
Underpinning
beliefs and rationale
Reflection
Students as
learners
Section 1
Influences on, and state of current mathematics curricula and pedagogical direction
Section 2
Influence of teachers’ belief and knowledge on teacher practice
Section 3
The enacted curriculum

12. THE RESEARCH RESEARCH QUESTIONS:
To gain insight into why and how secondary school mathematics teachers choose to use or choose not to use manipulatives in their classrooms, the research is guided by the following questions:
How do secondary school teachers view the use of manipulatives in teaching mathematics?
How do secondary school teachers describe their use of manipulatives in teaching mathematics?
DATA INSTRUMENT:
Data collected through semi-structured interviews with 8 (max.) secondary school mathematics teachers; with the key focus of:
teachers’ views and beliefs about mathematics, and mathematics teaching and learning
teachers’ views of the use of manipulatives for teaching mathematics
teachers’ descriptions of their use of manipulatives in the classroom.
Leading to the research question…
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I plan to collect data through one-hour interviews with up to 8 teachers. The number of participants is pragmatic in that it appears to be a reasonable balance between gaining different views and the limited time available in the study
The three main areas that guide the conversation are shown…

13. Key dimensions from research
DATA ANALYSIS
Key dimensions from research
Perceived value for all students
Math meaning behind their use
Uncertainty how to use them
Time and curriculum coverage pressures
Changes to class environment, control issues..
Support environment (peer, dept. admin.)
Logistics, availability, $$... …
The framework led to the general topics for discussion during the data collection process while the research provides some initial ‘top-level’ themes in the subsequent analysis.

14. Awaiting Ethics Approval
TIMELINE…
Awaiting Ethics Approval
Recruitment through open invitation to participate sent by COMA (and/or directed to willing candidates)
Data collection through interviews
Data analysis
Thesis
Now
End May
June
July
Fall
So, the timeline for the study is as follows:

15. Many thanks for your attention
Adrian Jones
Many thanks for your attention….