1) The two Montessori teachers interviewed believed that Montessori manipulatives are essential for teaching fractions and that their use equates to learning.
2) They distinguished Montessori education as superior to public school systems, feeling Montessori manipulatives better support mathematical understanding.
3) However, the teachers did not discuss challenges with fractions or how manipulatives specifically aid fraction instruction beyond working with the Montessori method.
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2. My interest in this topic
• Montessori educator
• Montessori head of school
• Struggled with mathematics at school
• “Found” Montessori
• Power of beliefs
3. Overview
• The problem
• Research on manipulatives
• Sociocultural theoretical framework
• Research design
• Findings
4. Teaching fractions
Fractions involve “relations between quantities” and one cannot count fractions as
there is “an infinite number of fractions between any two fractions or any two whole
numbers” (Lin, Becker, Ko, & Byun, 2013, p. 309).
Teachers may teach the way they have been taught (Pajares, 1992).
Teachers’ beliefs affect the way they teach (Negueruela-Azarola, 2011).
Arithmetic misconceptions learned in school are particularly resistant to change
(Green, Piel, & Flowers, 2008).
5. Definitions of manipulatives
• They can be moved around, rearranged, stacked (Uribe-Florez &Wilkins, 2010)
• They concretely represent abstract mathematical ideas (Moyer, 2001)
• They make abstract concepts salient (Zukerman, Arida, & Resnick, 2010)
• They enable a hands-on visual exploration of concepts (Chisnall & Maher, 2007)
6. Use of manipulatives
• Young children are expected to obtain cognitive benefits, such as abstract
reasoning, from exploring mathematical concepts with manipulatives
(Montessori, 1964).
• Efficacy of manipulatives in mathematics instruction is inconsistent (Carbonneau,
Marley, & Scott, 2013).
• Teachers’ attitudes towards the use of manipulatives is mixed as well (Green et al,
2008)
9. Vygotsky and Montessori’s similar ideas about learning
Development as multifaceted and
dynamic
• Social origins
• Mediation through tools
• Biological and cultural lines
• Consequences of activity
Development as multifaceted and
dynamic
• The prepared environment (culture)
• Development through work
• Biological and cultural lines (absorbent
mind)
• Consequences of activity
10. Allows for an investigation of teachers’
experiences as understood by them.
Recognizes that pedagogical practices are
a reflection the teacher, her educational
and personal history, her identity as a
teacher, and the context in which she
works, as well as the wider community
(Adler, 2001, p. 36).
Recognizes the importance of beliefs as
drivers of teachers’ actions (Negueruela-
Azarola, 2011).
A sociocultural
perspective:-
11. Research design
• Qualitative, using interviews
• Participants were two Montessori teachers – Brenda andAlicia
• Ethics approval granted, informed consent obtained
• Interviews were recorded and transcribed
• Analysis – categorization of data
12. Findings – 3 points
1) Teachers believe Montessori manipulatives are essential to teaching fractions
“It is a pretty brilliant material!”
Alicia
14. 3) Distinction between Montessori and non-Montessori
Montessori held in high regard pedagogically
Montessori manipulatives superior
Public system offers minimal, or basic, education
15. DISCUSSION
Teachers were strongly convinced of the value of Montessori manipulatives
in supporting their students’ mathematical understanding.
Manipulatives did not stand alone but worked in conjunction with the whole
Montessori pedagogy, which included stories and supplying a context.
Teachers did not comment on any challenges learning fractions.
Teachers did not discuss in what way manipulatives supported their teaching
of fractions.
Teachers were biased towards Montessori manipulatives and discarded
other, non-Montessori, manipulatives.
16. References
• Adler, J. (2001). Teaching mathematics in multilingual classrooms. Boston: Kluwer.
• Carbonneau, K., Marley, S., & Selig, J. (2013). A meta-analysis of the efficacy of teaching mathematics with concrete
manipulatives, Journal of Educational Psychology, 105(2), 380-400.
• Chisnall, N. & Maher, M. (2007). Montessori mathematics in early childhood education. Curriculum Matters, 3, 6-23.
• Green, M., Piel, J., & Flowers, C. (2008). Reversing education majors’ arithmetic misconceptions with short-term
instruction using manipulatives. TheJournal of Educational Research, 101(4), 234-242.
• Lin, C-Y., Becker, J., Ko,Y-Y., & Byun, M-R. (2013). Enhancing pre-service teachers’ fraction knowledge through open
approach instruction. The Journal of Mathematical Behaviour, 32, 309-330.
• Montessori, M. (1964). The Montessori Method, NewYork, NY: Schocken.
• Moyer, P. (2001). Are we having fun yet? How teachers use manipulatives to teach mathematics. Educational Studies in
Mathematics, 47, 175-197.
• Pajares, M. F. (1992).Teachers’ beliefs and educational research: Cleaning up a messy construct. Review of Education
Research, 62(3), 301-332.
• Uribe-Florez, L.J., &Wilkins, J.L. (2010). Elementary school teachers’ manipulative use. Science and Mathematics, 110(7),
363-371.
• Vygotsky, L. (1986). Thought and language, revised edition. Cambridge, MA:The MIT Press.
• Zuckerman,O., Arida, S., & Resnick, M. (2005). Extending tangible interfaces for education: Digital Montessori-inspired
manipulatives. CHI 2005 In: Proceedings of the SIGCHIConference on Human Factors in Computing Systems CHI’05, New
York, NY: ACM Press, 859-868.
Editor's Notes
Hi, I am Alison Goss and I am a PhD candidate from the University of Ottawa. What I am presenting here today is a very small study I undertook as part of a course requirement for my PhD. My topic is teaching fractions with Montessori manipulatives.
My interest in this topic arose out of experiences I have had in my life. I taught in a Montessori school for a number of years, and taught ages 6-9, and 9-12. I was also the head of a Montessori school. I am one of those people Bill Barton mentioned in his talk on Tuesday morning who apologize for not being “good” at math. Its hard not to see it as my fault. I found the Montessori approach to teaching and learning a supportive and stimulating environment. I situated this research in a Montessori setting since the use of manipulatives is considered a cornerstone of the Montessori epistemology. In my own teaching I had found the use of manipulatives enhanced my learning and the learning of students. I wished to understand other Montessori teachers’ beliefs and experiences using manipulatives when teaching fractions
I begin with a brief mention of the challenges
Many teachers find teaching fractions to their students a challenge. Teaching fractions can be difficult because fractions involve “relations between quantities” and one cannot count fractions as there is an “infinite number of fractions between any two fractions or any two whole numbers. Furthermore teachers often teach the way they have been taught and if they have had negative experiences with fractions they are likely to convey this anxiety to their students. They may have misconceptions about fractions and these misconceptions are particularly difficult to dislodge.
There are a variety of understandings with regards to what are mathematical manipulatives. Moyer’s (2001) definition that they concretely represent abstract mathematical ideas is most similar to the Montessori understanding of the function of manipulatives and that is to make abstract concepts salient and to enable hands-on visual exploration of concepts. There appears to be a common understanding in the literature that the goal of use of manipulatives is to offer students a way to understand abstract mathematical ideas by enabling them to connect to these ideas through concrete or physical materials.
Montessori manipulatives are designed to be attractive and appealing to the students. And to foster independent work, or work free from the teacher. There are other manipulatives for fraction work as well.
I used a sociocultural theoretical framework in this project based on the work of Vygotsky. For interest I discuss here ideas that I believe Montessori and Vygotsky shared in common although they may have been articulated differently.
My intention is not to force this comparison but I do believe they shared much in common in their theories of learning.
A sc perspective allows for an investigation of Montessori teachers’ experiences as understood by them in the context of their own practices. Specifically it allows for the forefronting of teachers’ beliefs as emerging from sense making tasks, and as powerful drivers of teachers’ actions.
As a Montessori teacher myself I have contact with a number of elementary Montessori teachers. I invited two teachers to take part in my research project. I contacted them via email and described the scope and purpose of the study. Both teachers agreed to take part in the study. Alicia and Brenda. The names are pseudonyms chosen by me. Each teacher has at least 10 years’ experience teaching Montessori at various age groupings of students. The teachers had not trained for, nor taught at public schools. The time for this study was limited, as was the availability of the teachers so I decided on a joint interview with the two teachers. I also believed that a joint interview would foster a good discussion between the teachers and this proved to be the case. Alicia and Brenda knew each other and were very amenable to the suggestion that they be interviewed together. The meeting with them lasted only 45 minutes as there was an unexpected family issue that Alicia had to deal with but this turned out to be enough time. For the analysis I transcribed the recorded interviews. Once comfortable with the contents of the transcription and when I had a general understanding the conversation I began to pay close attention to the specific words and phrases the teachers used. I noticed certain phrases and words were repeated which led me to understand that there were ideas that were important to the participants. I put these under various headings. Some headings had more entries than others, I continued to refine the headings to see if some were subheadings of others and thereby I managed my data. I considered each category in turn and attempted to make sense of what the teachers were referring to and what they understood these concepts to me.
It was clear that Brenda and Alicia held the manipulatives in high regard. Brenda saw the manipulatives as a critical part to teaching fractions. To her the value of the manipulatives lay in creating a visual representation of an abstract concept. She referred many times to visualization and explained how she “visualizes the materials” and she “saw the materials”. She explained how the students could extend their understanding of ratios to the “real world” because “they got a very good visual” These comments reflect what Moyer (2001) suggested the role of manipulatives is and that is to concretely represent abstract mathematical ideas.
For Brenda and Alicia the value of manipulative use went beyond helping students with abstract concepts. They believed the manipulatives essential and made learning come alive and that they support student understanding in mathematics. They believe this is done in part with manipulatives but also by way the Montessori pedagogy integrates a context, or creates a story with the mathematical teaching. For instance, Alicia talked about how there is a “story” integrated into the mathematics teaching of fractions. She explained that the word “fraction” is derived from the Latin fractere meaning to break. She explained how the stories connected the students to the past and to other people and that that is so much more interesting to the students. As the research literature showed, teachers’ attitudes towards manipulatives are an essential ingredient to their implementation (Green, et al, 2008).
It was apparent that the two participants made a strong distinction between Montessori schooling and non-Montessori schooling. They held the Montessori approach in much higher regard and academically superior. Alicia referred to a student who was struggling with fractions even though “she had gone through Montessori all the way” the implication being that this should not be the case. The two participants also distinguished between Montessori manipulatives and non-Montessori manipulatives. They called the Montessori fraction material “proper”, and “beautiful and lovely to hold”. The non-Montessori material was plastic, and stored in the “reject room”.
This small study reinforced my belief and expectation that Montessori teachers would value the use of manipulatives in the teaching of fractions. Both teachers seemed strongly convinced of the value of Montessori manipulatives to support their students mathematical understanding. What was new to me was their negative bias to non-Montessori manipulatives and schooling. Also of interest was Brenda’s repeated reference to visualization as a key concept facilitating students’ learning.
