Rationale for Year 6 Fractions Programme, listing references used in designing the programme.

1. Mathematics in Middle and Upper Primary
Fall
EDUC8505 08
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
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2. Mathematics in Middle and Upper Primary EDUC8505
& 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
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3. Mathematics in Middle and Upper Primary EDUC8505
between concepts and conventional language/symbols, enabling semiotic meaning
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

4. Mathematics in Middle and Upper Primary EDUC8505
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.
Retrieved from http://www.curriculum.wa.edu.au
Confrey, J. & Carrejo, D. (2005). Chapter 4: Ratio and Fraction: The Difference
Between Epistemological Complementarity and Conflict. Journal for
Research in Mathematics Education.
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5. Mathematics in Middle and Upper Primary EDUC8505
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

6. Mathematics in Middle and Upper Primary EDUC8505
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

7. Mathematics in Middle and Upper Primary EDUC8505
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:
10.1007/s10649-011-9306-5.
Sharon McCleary 7

8. Mathematics in Middle and Upper Primary EDUC8505
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

9. Mathematics in Middle and Upper Primary EDUC8505
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.
Retrieved from http://www.curriculum.wa.edu.au
Confrey, J. & Carrejo, D. (2005). Chapter 4: Ratio and Fraction: The Difference
Between Epistemological Complementarity and Conflict. Journal for
Research in Mathematics Education.
Sharon McCleary
5

10. Mathematics in Middle and Upper Primary EDUC8505
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

11. Mathematics in Middle and Upper Primary EDUC8505
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

12. Mathematics in Middle and Upper Primary EDUC8505
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:
10.1007/s10649-011-9306-5.
Sharon McCleary 12

13. Mathematics in Middle and Upper Primary EDUC8505
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