fractions rationale

Mathematics in Middle and Upper Primary EDUC8505

Rationale

Mathematics is a discipline which has evolved from the human need to measure and

communicate about time, quantity and space (Moursund, 2002). It is inherently

abstract, applicable over a wide field and uses symbols to represent mathematical

concepts.

Traditional theoretical frameworks associated with children’s mathematical thinking

include empiricism, where knowledge is external and acquired through the senses,

(neo)nativism/rationalism which emphasises the in-born capabilities of the child to

reason, and interactionalism, which recognises interacting roles of nature and

experience, and considers the child as active in knowledge construction (Lester, 2007).

A central part of each of these frameworks is experiences, which allow children to

internalise or express knowledge. Experiences provide opportunities to learn, which

are considered “the single most important predictor of student achievement” (National

Research Council, 2001, p334; cited in Lester, 2007), and allow children to acquire

physical, socio-conventional and logico-mathematical knowledge (Piaget, 1967, cited

in Kamii, 2004). They are instrumental in supporting student affect, which plays a

crucial role in mathematics teaching and learning (Hart&Walker, 1993, cited in

Baroody, 1998).

Experiences should illustrate a wide variety of examples relating to the key concept in

different contexts, to facilitate students forming multiple representations and

connections, and building conceptual understanding, rather than simply applying

procedural knowledge. Brownell (cited in Lester, 2007) refers to conceptual

understanding as mental connections among mathematical facts, procedures and

ideas. Vergnaud (1983, cited in Lester, 2007) introduced the concept of the

multiplicative conceptual field, a complex system of interrelated concepts, student

ideas (competencies and misconceptions), procedures, problems, representations,

objects, properties and relationships that cannot be studied in isolation, including

multiplication, division, fractions, ratios, simple and multiple proportions, rational

numbers, dimensional analysis and vector spaces. The programme provides varied

activities representing the five sub-constructs of fractions (Kieren, 1980, cited in Way

Sharon McCleary5

08Fall


& Bobis, 2011) part-whole, measure, quotient, operator, ratio), and encourages

connections between different rational numbers (e.g. Lesson 6: Fractions as quotients,

incorporating relational concepts and using numbers with common factors, which

support richer interconnections (Empson, 2005)).

Planned experiences should also aim to expose misconceptions, prevent the formation

of new ones (Bottle, 2005), and cater to students of different ability levels by using

open-ended activities. The programme achieves this by selecting activities that target

common misconceptions and require students to disprove them with concrete

materials (e.g. Lesson 3: Show Me A Half).

Concrete materials are central to assisting students in the concrete operations phase of

development understand mathematical concepts (Kamii, 2004). However their use

does not automatically result in mathematical learning, as students can focus on

unintended aspects and fail to abstract the intended concept (Gray, 1999).

Consequently, opportunities for exploring the manipulative before using it are given

(Lesson 4: Exploring Pattern Blocks), and use is closely aligned with conceptual

understanding and linked to symbolic conventions in order to promote purposeful

connections in students’ minds.

Several frameworks characterise mathematical learning as progressing from

physical/concrete interaction, to generalising abstract ideas/concepts and

representing them symbolically (Cowan, 2006; Lester, 2007; Baroody, 1998). Visual

imagery constructed from concrete experiences is central to this progression, and its

role in assisting learning has been addressed by several researchers, including

Presmeg, Goldin and Thomas. Consequently, several experiences in the programme

use concrete materials to encourage clear visual images that may assist children in

thinking mathematically (e.g. Lesson 1: water in glasses, Lessons 4&5: Pattern Blocks).

Correct mathematical language and writing conventions are also crucial to this

process, since effective communication is pivotal in clarifying inconsistencies between

the child’s inner understandings and correct conceptual understanding, and in

allowing opportunities for exchanges between peers, expanding strategy knowledge

through social learning (Vygotsky, 1978). The teacher’s role is to provide clear links

between concepts and conventional language/symbols, enabling semiotic meaning

Sharon McCleary 2


making without stifling inherent thought processes. Opportunities to build fluency are

provided in each lesson of the programme through reading, writing, talking and

listening.

Another feature linked to developing conceptual understanding is allowing students to

actively expend effort in making sense of important mathematical ideas. Festinger’s

(1957) theory of cognitive dissonance describes perplexity as a central impetus for

cognitive growth, and Hatano (1988) identifies cognitive incongruity as the critical

trigger for developing reasoning skills that display conceptual understanding (Lester,

2007). This is consistent with constructivist ideas of presenting problems near the

boundary of the student’s Zone of Proximal Development (Vygotsky, 1978), allowing

sufficient challenge to promote thinking and application of conceptual knowledge,

while supporting opportunities for success and maintaining positive affect: “Acquired

knowledge is most useful to a learner when it is discovered through their own

cognitive efforts, related to and used in reference to what one has known before”

(Bruner, cited in Cowan, 2006, pg 26). The programme incorporates problem solving

allowing different solution methods: Lessons 6 & 12 provide additive and

multiplicative thinking arising from invented strategies for division questions,

allowing students opportunities to ‘struggle’ with relevant mathematical concepts in

authentic scenarios.

The teacher’s role encompasses providing children with engaging, challenging and

enjoyable experiences which emphasise conceptual understanding and promote a

positive attitude towards mathematics. Implicit in this is creating a classroom

environment which allows opportunities for discussion, assists students in becoming

fluent with conventional mathematical language/symbols and is accepting of invented

strategies and solution methods. This facilitates students’ forming connections

between multiple representations and abstracting meaning from experiences to

progress and apply their mathematical thinking. (810 words)

Sharon McCleary 3


References

Baroody, A. & Coslick, R. (1998). Fostering Children’s Mathematical Power, An

Investigative Approach to K-8 Mathematics Instruction. Lawrence Erlbaum

Associates, London.

Bottle, G. (2005). Teaching Mathematics in the Primary School. Continuum,

London.

Burns, M. (2001). Lessons for Introducing Fractions. Math Solution Publications.

California.

Cathcart, W., Pothier, Y., Vance, J. & Bezuk, N. (2011). Learning Mathematics in

Elementary and Middle Schools, A Learner-Centered Approach. 5th Edition.

Pearson Education, Boston.

Clarke, D. & Roche, A. (2009). Students’ fraction comparison strategies as a

window into robust understanding and possible pointers for instruction.

DOI: 10.1007/s10649-009-9198-9.

Curriculum Council (Ed.). (1998). Curriculum Framework, Kindergarten to Year 12

Education in Western Australia (Mathematics Learning

Area Statement). Curriculum Council of Western Australia. Perth. WA.

Retrieved from http://www.curriculum.wa.edu.au

Curriculum Council. (2005). Outcomes and Standards Framework and

Syllabus Documents, Progress Maps and Curriculum Guide. Curriculum

Council of Western Australia. Perth. WA.


Confrey, J. & Carrejo, D. (2005). Chapter 4: Ratio and Fraction: The Difference

Between Epistemological Complementarity and Conflict. Journal for

Research in Mathematics Education.

Sharon McCleary5

http://www.curriculum.wa.edu.au/



Copeland, R. (1970). How Children Learn Mathematics, Teaching Implications of

Piaget’s Research, The Macmillan Company, London.

Cowan, P. (2006). Teaching Mathematics, A Handbook for Primary & Secondary

School Teachers, Routledge, New York.

Department of Education Victoria. (2011). Fractions and Decimals Online Interview

Classroom Activities. Retrieved from http://www.education.vic.gov.au

Department of Education and Training Western Australia. (2007). Middle Childhood:

Mathematics/ Number Scope and Sequence. Retrieved from:

http://www.curriculum.wa.edu.au/internet/Years_K10/Curriculum_Resources

Devlin, K. (2006). Mathematical Association of America, How do we learn math?.

Retrieved from: www.maa.org/devlin/devlin_03_06.html

Dienes, Z.P. (1973). Mathematics through the senses, games, dance and art,

NFER Publishing Company, Ltd, New York.

Downton, A., Knight, R., Clarke, D. & Lewis, G. (2006). Mathematics Assessment

for Learning: Rich Tasks & Work Samples. Mathematics Teaching and

Learning Centre. Melbourne. Australia.

Empson, S.B., Junk, D., Dominguez, H., & Turner, E. (2005). Fractions as the co-

ordination of multiplicatively related quantities: a cross-sectional study of

children’s thinking. Educational Studies in Mathematics 63, pg1-28.

Flewelling, G., Lind, J. & Sauer, R. (2010). Rich Learning Tasks in Number. The

Australian Association of Mathematics Teachers. South Australia.

Fraser, C. (2004). The development of the common fraction concept in Grade 3

learners. Pythagoras 59. pg26-33.

Gray, E., Pitta, D. & Tall, D. (1999). Objects, Actions and Images: A Perspective on

Early Number Development. Mathematics Education Research Centre,

Coventry, UK.

Sharon McCleary 5

http://www.education.vic.gov.au/


Halberda, J. & Feigenson, L. (2008). Developmental Change in the Acuity of the

“Number Sense”: The Approximate Number System in 3-, 4-, 5-, and 6-Year-

Olds and Adult. Developmental Psychology, Vol. 44, No. 5, pg 1457-1465.

Kamii, C. (1984). Autonomy as the aim of childhood education: A Piagetian

Approach, Galesburg, IL.

Kamii, C. (2004). Young Children Continue to Reinvent Arithmetic – 2nd Grade-

Implications of Piaget’s Theory, 2nd Edition, Teachers College Press, London.

Lappan, G., Fey, J., Fitzgerald, W., Friel, S & Phillips, E. (2002). Bits and Pieces I

Understanding Rational Numbers. Prentice Hall, Illinois.

Lester, F. (Ed.) (2007). Second Handbook of Research on Mathematics Teaching

and Learning. National Council of Teachers of Mathematics, USA.

McClure, L. (2005). Raising the Profile, Whole School Maths Activities for Primary

Pupils. The Mathematical Association. Leicester.

McIntosh, A., Reys, B., Reys, R. & Hope, J. (1997). NumberSENSE: Simple Effect

Number Sense Experiences, Dale Seymour Publications, USA.

Moseley B. (2005). Students’ Early Mathematical Representation Knowledge: The

Effects of Emphasizing Single or Multiple Perspectives of the Rational

Number Domain in Problem Solving.

Moss, J. & Case, R. (1999). Developing Children’s Understanding of the Rational

Numbers: A New Model and an Experimental Curriculum. Journal for

Research in Mathematics Education. Vol. 30. No. 2. Pp122-47.

Moursand, D., (2006), Mathematics, Retrieved from:

http://darkwing.uoregon.edu/~moursund/math/mathematics.htm

Muir, T. (2008). Principles of Practice and Teacher Actions: Influences on Effective

Teaching of Numeracy. Mathematics Education Research Journal. Vol. 20,

No. 3, pg 78-101.

Nunes, T. & Bryant, P. (1996). Children Doing Mathematics. Blackwell Publishers,

Massachusetts, USA.

Presmeg, N. (n.d.). Research on Visualisation in Learning and Teaching

Sharon McCleary 6


Mathematics, Illinois State University.

Radford, L., Schubring, G. & Seeger, F. (2011). Signifying and meaning-making in

mathematical thinking, teaching an learning.

DOI: 10.1007/s10649-011-9322-5.

Reys, R. & Yang, D.C. (1998). Relationship Between Computational Performance

and Number Sense Among Sixth- and Eighth-Grade Students in Taiwan.

Journal for Research in Mathematics Education. Vol. 29, No. 2, pg 225-237.

Schneider, M., Grabner, R. & Paetsch, J. (2009). Mental Number Line, Number

Line Estimation, and Mathematical Achievement: Their Interrelations in

Grads 5 and 6. Journal of Educational Psychology. Vol. 101, No. 2. pgs 359-

372.

Siegler, R., Thompson, C. & Schneider, M. (2011). An Integrated theory of whole

number and fractions development. Cognitive Psychology. Vol. 62. pp273-

296.

Smith, C., Solomon, G. & Carey, S. (2005). Never getting to zero: elementary

school students’ understanding of the infinite divisibility of number and

matter. Cognitive Psychology. Vol.51. pp101-140.

Stenmark, J. & Bush, W. (2001). Mathematics Assessment, A Practical Handbook.

National Council of Teachers of Mathematics. VA.

Sullivan, P. & Lilburn, P. (2004). Open-ended Maths Activities, Using ‘good’

questions to enhance learning in Mathematics. 2nd Edition, Oxford University

Press, Oxford.

The Australian Curriculum-Mathematics, Version 1.1, (2010). Australian

Curriculum, Assessment and Reporting Authority [ACARA], Retrieved from:

http://www.australiancurriculum.edu.au

Vygotsky, L., (1978). Mind in Society, Harvard University Press, Cambridge, MA.

Way, J. & Bobis, J. (2011). Fractions, Teaching for Understanding. The Australian

Association of Mathematics Teachers Inc. South Australia.

Wilkerson-Jerde, M. & Wilensky, U. (2011). How do mathematicians learn math?:

resources and acts for constructing and understanding mathematicians, doi:

Sharon McCleary 7

http://www.australiancurriculum.edu.au/


10.1007/s10649-011-9306-5.

Willis, S., Devlin, W., Jacob, L., Powell, B., Tomazos, D. & Treacy, K. (2004),

First Steps in Mathematics: Number (Book 1). Rigby. Australia.

Sharon McCleary 8


References

Baroody, A. & Coslick, R. (1998). Fostering Children’s Mathematical Power, An

Investigative Approach to K-8 Mathematics Instruction. Lawrence Erlbaum

Associates, London.

Bottle, G. (2005). Teaching Mathematics in the Primary School. Continuum,

London.

Burns, M. (2001). Lessons for Introducing Fractions. Math Solution Publications.

California.

Cathcart, W., Pothier, Y., Vance, J. & Bezuk, N. (2011). Learning Mathematics in

Elementary and Middle Schools, A Learner-Centered Approach. 5th Edition.

Pearson Education, Boston.

Clarke, D. & Roche, A. (2009). Students’ fraction comparison strategies as a

window into robust understanding and possible pointers for instruction.

DOI: 10.1007/s10649-009-9198-9.

Curriculum Council (Ed.). (1998). Curriculum Framework, Kindergarten to Year 12

Education in Western Australia (Mathematics Learning

Area Statement). Curriculum Council of Western Australia. Perth. WA.


Curriculum Council. (2005). Outcomes and Standards Framework and

Syllabus Documents, Progress Maps and Curriculum Guide. Curriculum

Council of Western Australia. Perth. WA.


Confrey, J. & Carrejo, D. (2005). Chapter 4: Ratio and Fraction: The Difference

Between Epistemological Complementarity and Conflict. Journal for

Research in Mathematics Education.

Sharon McCleary5




Copeland, R. (1970). How Children Learn Mathematics, Teaching Implications of

Piaget’s Research, The Macmillan Company, London.

Cowan, P. (2006). Teaching Mathematics, A Handbook for Primary & Secondary

School Teachers, Routledge, New York.

Department of Education Victoria. (2011). Fractions and Decimals Online Interview

Classroom Activities. Retrieved from http://www.education.vic.gov.au

Department of Education and Training Western Australia. (2007). Middle Childhood:

Mathematics/ Number Scope and Sequence. Retrieved from:

http://www.curriculum.wa.edu.au/internet/Years_K10/Curriculum_Resources

Devlin, K. (2006). Mathematical Association of America, How do we learn math?.

Retrieved from: www.maa.org/devlin/devlin_03_06.html

Dienes, Z.P. (1973). Mathematics through the senses, games, dance and art,

NFER Publishing Company, Ltd, New York.

Downton, A., Knight, R., Clarke, D. & Lewis, G. (2006). Mathematics Assessment

for Learning: Rich Tasks & Work Samples. Mathematics Teaching and

Learning Centre. Melbourne. Australia.

Empson, S.B., Junk, D., Dominguez, H., & Turner, E. (2005). Fractions as the co-

ordination of multiplicatively related quantities: a cross-sectional study of

children’s thinking. Educational Studies in Mathematics 63, pg1-28.

Flewelling, G., Lind, J. & Sauer, R. (2010). Rich Learning Tasks in Number. The

Australian Association of Mathematics Teachers. South Australia.

Fraser, C. (2004). The development of the common fraction concept in Grade 3

learners. Pythagoras 59. pg26-33.

Gray, E., Pitta, D. & Tall, D. (1999). Objects, Actions and Images: A Perspective on

Early Number Development. Mathematics Education Research Centre,

Coventry, UK.

Sharon McCleary 10

http://www.education.vic.gov.au/


Halberda, J. & Feigenson, L. (2008). Developmental Change in the Acuity of the

“Number Sense”: The Approximate Number System in 3-, 4-, 5-, and 6-Year-

Olds and Adult. Developmental Psychology, Vol. 44, No. 5, pg 1457-1465.

Kamii, C. (1984). Autonomy as the aim of childhood education: A Piagetian

Approach, Galesburg, IL.

Kamii, C. (2004). Young Children Continue to Reinvent Arithmetic – 2nd Grade-

Implications of Piaget’s Theory, 2nd Edition, Teachers College Press, London.

Lappan, G., Fey, J., Fitzgerald, W., Friel, S & Phillips, E. (2002). Bits and Pieces I

Understanding Rational Numbers. Prentice Hall, Illinois.

Lester, F. (Ed.) (2007). Second Handbook of Research on Mathematics Teaching

and Learning. National Council of Teachers of Mathematics, USA.

McClure, L. (2005). Raising the Profile, Whole School Maths Activities for Primary

Pupils. The Mathematical Association. Leicester.

McIntosh, A., Reys, B., Reys, R. & Hope, J. (1997). NumberSENSE: Simple Effect

Number Sense Experiences, Dale Seymour Publications, USA.

Moseley B. (2005). Students’ Early Mathematical Representation Knowledge: The

Effects of Emphasizing Single or Multiple Perspectives of the Rational

Number Domain in Problem Solving.

Moss, J. & Case, R. (1999). Developing Children’s Understanding of the Rational

Numbers: A New Model and an Experimental Curriculum. Journal for

Research in Mathematics Education. Vol. 30. No. 2. Pp122-47.

Moursand, D., (2006), Mathematics, Retrieved from:

http://darkwing.uoregon.edu/~moursund/math/mathematics.htm

Muir, T. (2008). Principles of Practice and Teacher Actions: Influences on Effective

Teaching of Numeracy. Mathematics Education Research Journal. Vol. 20,

No. 3, pg 78-101.

Nunes, T. & Bryant, P. (1996). Children Doing Mathematics. Blackwell Publishers,

Massachusetts, USA.

Presmeg, N. (n.d.). Research on Visualisation in Learning and Teaching

Sharon McCleary 11


Mathematics, Illinois State University.

Radford, L., Schubring, G. & Seeger, F. (2011). Signifying and meaning-making in

mathematical thinking, teaching an learning.

DOI: 10.1007/s10649-011-9322-5.

Reys, R. & Yang, D.C. (1998). Relationship Between Computational Performance

and Number Sense Among Sixth- and Eighth-Grade Students in Taiwan.

Journal for Research in Mathematics Education. Vol. 29, No. 2, pg 225-237.

Schneider, M., Grabner, R. & Paetsch, J. (2009). Mental Number Line, Number

Line Estimation, and Mathematical Achievement: Their Interrelations in

Grads 5 and 6. Journal of Educational Psychology. Vol. 101, No. 2. pgs 359-

372.

Siegler, R., Thompson, C. & Schneider, M. (2011). An Integrated theory of whole

number and fractions development. Cognitive Psychology. Vol. 62. pp273-

296.

Smith, C., Solomon, G. & Carey, S. (2005). Never getting to zero: elementary

school students’ understanding of the infinite divisibility of number and

matter. Cognitive Psychology. Vol.51. pp101-140.

Stenmark, J. & Bush, W. (2001). Mathematics Assessment, A Practical Handbook.

National Council of Teachers of Mathematics. VA.

Sullivan, P. & Lilburn, P. (2004). Open-ended Maths Activities, Using ‘good’

questions to enhance learning in Mathematics. 2nd Edition, Oxford University

Press, Oxford.

The Australian Curriculum-Mathematics, Version 1.1, (2010). Australian

Curriculum, Assessment and Reporting Authority [ACARA], Retrieved from:

http://www.australiancurriculum.edu.au

Vygotsky, L., (1978). Mind in Society, Harvard University Press, Cambridge, MA.

Way, J. & Bobis, J. (2011). Fractions, Teaching for Understanding. The Australian

Association of Mathematics Teachers Inc. South Australia.

Wilkerson-Jerde, M. & Wilensky, U. (2011). How do mathematicians learn math?:

resources and acts for constructing and understanding mathematicians, doi:

Sharon McCleary 12

http://www.australiancurriculum.edu.au/


10.1007/s10649-011-9306-5.

Willis, S., Devlin, W., Jacob, L., Powell, B., Tomazos, D. & Treacy, K. (2004),

First Steps in Mathematics: Number (Book 1). Rigby. Australia.

Sharon McCleary 13