TEACHING FRACTIONS WITH MANIPULATIVES

Two Montessori teachers’ experiences

Alison Goss, PhD candidate,University of Ottawa, Canada.

My interest in this topic• Montessori educator• Montessori head of school• Struggled with mathematics at school• “Found” Montessori• Power of beliefs

Overview• The problem• Research on manipulatives• Sociocultural theoretical framework• Research design• Findings

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).

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)

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)

SOCIOCULTURAL THEORETICAL FRAMEWORK

Vygotsky and Montessori –Some shared ideas

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

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:-

Research design

• Qualitative, using interviews• Participants were two Montessori teachers – Brenda and Alicia• Ethics approval granted, informed consent obtained• Interviews were recorded and transcribed• Analysis – categorization of data

Findings – 3 points

1) Teachers believe Montessori manipulatives are essential to teaching fractions

“It is a pretty brilliant material!”

Alicia

2) Manipulative use is equated with learning

“Learning comes alive” Alicia

3) Distinction between Montessori and non-Montessori

Montessori held in high regard pedagogically

Montessori manipulatives superior

Public system offers minimal, or basic, education

DISCUSSIONTeachers 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.

Refer en ces

• 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. The Journal 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, New York, 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 SIGCHI Conference on Human Factors in Computing Systems CHI’05, New York, NY: ACM Press, 859-868.